Issue |
A&A
Volume 509, January 2010
|
|
---|---|---|
Article Number | A84 | |
Number of page(s) | 8 | |
Section | Astrophysical processes | |
DOI | https://doi.org/10.1051/0004-6361/200810780 | |
Published online | 22 January 2010 |
Accuracy of spectroscopy-based radioactive dating of stars
H.-G. Ludwig1,2 - E. Caffau2 - M. Steffen3 - P. Bonifacio1,2,4 - L. Sbordone1,2
1 - CIFIST Marie Curie Excellence Team, France
2 - GEPI, Observatoire de Paris, CNRS, Université Paris Diderot, Place
Jules Janssen, 92190 Meudon, France
3 - Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482
Potsdam, Germany
4 - Istituto Nazionale di Astrofisica, Osservatorio Astronomico di
Trieste, via Tiepolo 11, 34143 Trieste, Italy
Received 10 August 2009 / Accepted 7 October 2009
Abstract
Context. Combined spectroscopic abundance analyses
of stable and radioactive elements can be applied for deriving stellar
ages. The achievable precision depends on factors related to
spectroscopy, nucleosynthesis, and chemical evolution.
Aims. We quantify the uncertainties arising from the
spectroscopic analysis, and compare these to the other error sources.
Methods. We derive formulae for the age
uncertainties arising from the spectroscopic abundance analysis, and
apply them to spectroscopic and nucleosynthetic data compiled from the
literature for the Sun and metal-poor stars.
Results. We obtained ready-to-use analytic formulae
of the age uncertainty for the cases of stable+unstable and
unstable+unstable chronometer pairs, and discuss the optimal relation
between to-be-measured age and mean lifetime of a radioactive species.
Application to the literature data indicates that, for a single star,
the achievable spectroscopic accuracy is limited to about 20% for the
foreseeable future. At present, theoretical uncertainties in
nucleosynthesis and chemical evolution models form the precision
bottleneck. For stellar clusters, isochrone fitting provides a higher
accuracy than radioactive dating, but radioactive dating becomes
competitive when applied to many cluster members simultaneously,
reducing the statistical errors by a factor
.
Conclusions. Spectroscopy-based radioactive stellar
dating would benefit from improvements in the theoretical understanding
of nucleosynthesis and chemical evolution. Its application to clusters
can provide strong constraints for nucleosynthetic models.
Key words: Sun: abundances - stars: abundances - stars: fundamental parameters - line: formation
1 Introduction
The determination of absolute ages for stars is of paramount importance in astrophysics since ages allow the timescales on which stellar populations and their chemical composition evolve to be established. The presence of long-lived unstable elements in stellar atmospheres offers a way to date stars by investigating their radioactive decay spectroscopically. In its simplest form, one determines the photospheric abundance of a pair of elements, of which one is unstable. By means of theoretical modeling of the nucleosynthetic production channels for the two elements, the original abundance of the unstable nucleus can be estimated from the abundance of the stable one. This allows the time-span over which the unstable element decayed to be determined.
The idea emerged early after the discovery of the rapid neutron capture process (short r-process) as the source of production of radionuclides such as thorium and uranium (see Cowan et al. 1991, and references therein for an account of its early applications). Since then it has been applied in a number of cases (among the most recent Frebel et al. 2007) but, typically, as a ``side product'' of analyses of the more general n-capture enrichment patterns. To our knowledge, the only effort in recent years to exploit radioactive dating to systematically derive the age of a significant sample of stars has been the one by del Peloso et al. (2005a,b,c).
One interest in such dating techniques lies in its applicability to field stars for which no distance information, hence no accurate information on surface gravity or absolute luminosity, is available, so that they cannot be placed with confidence on theoretical isochrones in the Hertzsprung-Russell diagram (HRD). For cluster stellar populations, on the other hand, HRD fitting usually provides rather precise absolute ages (e.g. Vandenberg et al. 1996). However, for stellar clusters (especially globular clusters), radioactive dating could produce an age estimate of comparable or even higher precision if applied to a large enough sample of stars, and would as such be very valuable as a test of the theoretical stellar isochrones.
Although potentially effective, spectroscopy-based radioactive dating is difficult to apply. The most easily measured long-lived radioactive element, thorium, is only accessible (with the exception of strongly r-process enriched stars) through a single, weak line in the optical spectrum of stars, in which high accuracy is needed in the measurement to provide a reliable age. Besides lying on the red wing of a Fe-Ni blend, this unique Th II line is itself also blended by weaker lines of vanadium and cobalt. The relative strength of the various components for the Sun are discussed in Caffau et al. (2008) where the line-list of del Peloso et al. (2005a) was applied. Although rather easily measured, the most frequently used reference element, europium, is believed to constitute a less-than-optimal reference. In fact, its nucleosynthetic ties with Th are rather weak, and the production ratio is sensitive to the originating supernova characteristics (Kratz et al. 2007; Farouqi et al. 2008).
Besides the astrophysical problem that the r-process site has not been unambiguously identified, it is unclear how universal the resulting abundance pattern of the r-process nucleosynthesis actually is. In view of the complex nuclear reactions shaping the r-process it is not obvious that the outcome is the same for a potentially wide range of the physical conditions governing the formation site(s) (Roederer et al. 2009; Arnould et al. 2007; Goriely & Clerbaux 1999; Sneden et al. 2008). To minimize the influence of poorly constrained physical conditions at the formation site it is desirable to use a reference element with an atomic mass as similar as possible to thorium. This should reduce the uncertainties in the formation properties to the uncertainties related to the properties of the involved nuclei as such, and environmental factors should cancel out. As pointed out before by various authors (e.g., Goriely & Clerbaux 1999) the Th/U pair is in this sense optimal, however, again difficult to apply since uranium is difficult to detect spectroscopically. Recently, Kratz et al. (2007) suggested the element hafnium as suitable reference element for thorium-based radioactive dating. Hafnium is closer in mass to thorium than europium, but more importantly, Kratz and collaborators showed by detailed modeling that the production ratio Th/Hf is not strongly varying over a wide range of neutron fluxes during the r-process. Since hafnium is a spectroscopically accessible element this makes it an interesting candidate as reference element.
Observationally, the forthcoming introduction of extremely large telescopes (ELTs) is likely to make radioactive dating of stars a much more exploitable technique, since the high signal-to-noise (S/N) ratio required is one of the main difficulties associated with such studies. In this perspective, a precise assessment of the uncertainties associated with such technique is currently lacking, and attempted here. We derive the formulae for the precision to which spectroscopy can provide age estimates. They are subsequently applied to abundance analyses of stars compiled from the literature. This paper is a ``spin-off'' of our work (Caffau et al. 2008) on the thorium and hafnium abundance in the Sun, and the Th/Hf chronometer pair often serves as an example in the following sections.
2 Formulae related to radioactive dating
2.1 Age uncertainties
Any sample of radioactive nuclei decays according to the exponential
law:
where nX0 is the number of nuclei at t=0, and nX the number of remaining nuclei at time t;

Knowing the initial and present number of nuclei, one can obtain the time interval that has elapsed since t=0. In practice, to use the decay processes to determine an age, one measures the number of nuclei of the unstable species (nX) relative to the number of nuclei of a stable element which is related to nX0. This is, e.g., the case for the Th/Hf pair. Let us call nR the number of nuclei of the stable reference species related to nX0. Let us further assume that the two nuclei have a common nucleosynthetic origin and that the relation between the initial number of nuclei nX0 and nR is given by
where

All astrophysically relevant unstable elements are trace species. Let us therefore suppose that the lines corresponding to the transitions we are interested in, are weak. In this case, the relation between equivalent width (W hereafter) and abundance (n) is linear:
C being a constant defined by the line transition under consideration (dependent on oscillator strength and excitation potential) and the thermal structure of the stellar atmosphere under investigation.
Many lines - including the Th II
resonance line - are blended with other species, and in practice the
abundance is obtained by fitting a synthetic profile. In this case the
equivalent width is obtained theoretically from the best-fitting
profile and the underlying abundances. The error on the equivalent
width may then be obtained by propagating the error in Eq. (5). One may also try
to derive an equivalent width for the blended line by subtracting
the theoretical equivalent width of the blending components from the
total W of the blend. However, in general, this
ignores saturation effects, and thus the ``corrected'' W
differs from the true one. As an example we may consider the Th II
resonance line in the Sun. Let us consider the theoretical computation
of Caffau
et al. (2008) based on a 3D
model: the W of the blend Th II+Co
I+V II, for an
assumed abundance
is 0.74 pm, the W of just the Co I+V
II blend is 0.31 pm, while
pm.
In spite of the fact that all the lines involved are very weak, it is
clear that the ``corrected'' W (0.43 pm)
differs from W(Th), due to saturation effects. Thus
one should always use the equivalent width of the line, rather than the
``corrected'' W, although in many cases the difference is so small that
it may be safely ignored. Alternatively, one may work with
Eq. (9)
and use the obtained abundances directly. In this case one has to
establish uncertainties by Monte-Carlo simulations.
Substituting Eqs. (3)
and (5) into
Eq. (2)
we obtain
The achievable precision in t is hence limited by the uncertainty in the knowledge of

From the standard formula for the propagation of uncorrelated
errors
we get for the variance of t
This equation is valid under the presumption that the errors of

Taking the Sun and the Th/Hf pair as an example, we have
(see Hf II
409.3 nm in
Table 3
in Caffau
et al. 2008), and
(see
Table 4 in Caffau
et al. 2008). Since 232Th
has a mean lifetime of
Gyr,
we obtain from Eq. (7)
under the most
optimistic conditions, assuming
and
,
an
error in the age determination of the order of
Gyr.
Even this rather limited precision is probably beyond the present state-of-the-art in stellar spectroscopy. However it is not implausible to reach a precision of the order of 2 Gyr in the future. The uncertainties due to the S/N in the spectrum may be considerably reduced by the next generation of ELTs, and uncertainties in the blending lines can be reduced through laboratory measurements and/or theoretical calculations. The precision currently attainable on the dating of Globular Clusters, using Main Sequence fitting, is a factor of two better (about 1 Gyr), however it may not be applied to field stars. For field stars which are evolved off the Main Sequence, if the distance is known, an age can be derived through comparison with theoretical isochrones. However, distances are difficult to measure and a small error in the distance propagates to a great error in age. In principle, radioactive dating has the potential to provide ages of individual field stars, even if their distances are not precisely known.
Unfortunately, in the literature equivalent widths and their
uncertainties are
usually not given so that a direct application of the previous formulae
is
not possible. In Sect. 2.2
we will see that they are nevertheless
useful to derive a statement of the optimal decay time of a species for
measuring a particular time interval. For now, we proceed by listing
the
previous relations in terms of the logarithmic abundances
.
Equation (4)
for the age
becomes
with the corresponding uncertainty
![$\ensuremath{{\rm Cor}\left[.,.\right]} $](/articles/aa/full_html/2010/01/aa10780-08/img84.png)

![]() |
(10) |
Equation (9) now explicitly contains the correlation among the measurements of the abundances of the two involved species. If abundance errors are positively correlated - as discussed previously and the Th/Hf pair serving as example - this can lead to a substantial reduction of the overall error with respect to the uncorrelated case. However, care has to be exercised when applying the formulae to literature data (as we intend to do). Quoted abundance errors are often purely statistical, uncorrelated uncertainties while correlated uncertainties related to atmospheric models are specified separately by providing sensitivities against changes of the stellar parameters.
2.2 Optimal decay times
We now discuss how the achievable precision depends on the time
interval
t we want to measure and the decay time of the
radioactive species, .
From Eq. (7)
or (9) it
might appear that a radioactive species with short
mean lifetime
provides the highest precision on t. However, this
reasoning ignores the fact that a rapid decay (
)
makes it more
difficult to measure WX
precisely. To make this connection manifest
we eliminate WX
and rewrite Eq. (7)
as
where we have used the relations
and
Equation (12) is obtained by combining Eqs. (1), (3), and (5), while Eq. (13) is a consequence of Cayrel's formula (Cayrel, 1988) which, for a spectrum that is photon noise dominated, states that the uncertainty of the W measurement






For brevity, we introduced an auxiliary function F of argument





![]() |
Figure 1:
The function |
Open with DEXTER |
We finally consider radioactive dating when measuring two
lines from two transitions of two unstable elements
which we call X and Y.
There are two decay laws to be considered for the two elements:
![]() |
(15) |
We suppose that


![]() |
(16) |
once

We then find the same result for


In practice, the only two radioactive nuclei which have been
used to date
stars are 232Th and 238U
with mean lifetimes of 20.3 Gyr and
6.5 Gyr, respectively. From Eq. (17) the effective
for
the Th/U pair amounts to 9.6 Gyr. From Fig. 1 it is clear that
the Th/U pair as well as U/R pairs (R for any stable
r-process element) are
well suited for measuring great ages around 12 Gyr, with Th/R
being
competitive depending on the actual value of the p
parameter. Towards ages
of a few Gyr the sensitivity of U/R pairs is about 2 to 3 times higher
than
the one of Th/R pairs, again depending on p. We
repeat that only the decay
times enter the considerations here; it is usually more difficult
to measure U than Th abundances so that the advantage of the shorter
life time
of U is offset. Finally, we ask whether there are other suitable
naturally
occurring radioactive elements with lifetimes even more favorable for
measuring
ages in the range of a few Gyr. The answer is ``no'': the lifetimes are
either
much too short or much too long - besides further complications like
exceedingly low abundances of the respective isotopes or the
availability of
suitable spectral lines.
3 Ages of metal-poor stars
As an application of the formulae we derived for the errors on age, we
consider metal-poor stars for which the thorium abundance has been
measured.
For all these stars the abundance of at least one other r-process
element is
available (see Table A.1).
Using various production ratios available in the literature
(see Table A.2),
we computed the ages of
these stars using Eq. (8)
and their uncertainties with
Eq. (9).
Correlations were neglected; correlations due to
uncertainties in stellar parameters are usually not included in the
error
budget, and correlations emerging from the data reduction (e.g.
background
reduction) not specified by the authors. Our results are summarized in
Table 1.
For star HE 1523-0901 the abundances of the single
elements were not available, but in Frebel
et al. (2007) abundance ratios are
reported, and measurement errors on the individual abundances which is
sufficient for our purposes. By using different production ratios we
intended
to separate the systematic uncertainties related to nucleosynthesis and
chemical evolution from the spectroscopic uncertainties as such. Since
the
production ratios are treated separately here their related uncertainty
in the
evaluation of
with Eq. (9)
is set to zero,
.
Table 1:
Ages and age uncertainties
of metal-poor stars and the
Sun obtained for different chronometer
pairs X/Y.
We like to point out that we applied the spectroscopy-related
uncertainties of
the abundances as given by the authors. There is no general consensus
about
the appropriate way to estimate the error on abundances, and the
approach
varies from author to author. For those elements which have several
measurable
lines the error can be estimated by looking at the rms line-to-line
abundance
scatter. One further might divide this error by
where N is the
number of lines, provided the errors on each line are independent.
However,
this assumption is often questionable, since in most cases the lines
are
measured from the same spectrum. Any error which affects the whole
spectrum
(e.g. an error in the background subtraction) will affect all the lines
in the
same (or at least in a similar) way. We thus argue that it is more
realistic
and conservative not to invoke this factor 1/
.
The ages of the individual metal-poor stars in Table 1 show a great
spread when considering different r-production
ratios, ,
and chronometer pairs. Also the age uncertainties stemming from the
spectroscopically determined
abundances,
,
are substantial. We note that our spectroscopy-related uncertainties
are commonly greater than the estimates given by the authors
themselves.
Figures 2
and 3
provide a graphical representation
of the situation for the stars CS 22892-052 and
CS 31082-001, respectively. We
picked these objects since spectroscopic measurements for a greater
number of
chronometer pairs exist. CS 31082-001 is of particular
interest since it is
strongly r-process element enhanced allowing the
measurement of the Th
abundance exceptionally from several lines making it more reliable.
For CS 22892-052 we left out the age estimates with the
highest uncertainties,
stemming from Th/Os and Th/Ir, since they were later improved
substantially. Both figures
illustrate that the dispersion due to different production ratios is
often
greater than the dispersion due to abundance uncertainties. This is
even the
case if one restricts the comparison to production ratios originating
from the
same author. As already pointed out in the original paper on the
abundance
analysis of CS 31082-001 by Hill et al. (2002)
the age derived from the Th/Eu pair
is obviously unrealistic despite rather accurate abundance
measurements. The
same holds to lesser extend for the Th/Ir pair. This points to
shortcomings in
our understanding of the r-process production
ratios and/or the chemical
history of the star. For CS 22892-052 all the reference elements except
Eu
lead to exceedingly great ages. One might speculate that the
Th/Hf age might be enlarged by an s-process
contribution to the Hf abundance:
although the star has a very low metalicity, it is also carbon
enhanced, so
that one cannot rule out the possibility that it may have a (so far
undetected) companion which went through the AGB phase, polluting the
photosphere of the primary. However, even this hypothesis cannot be
invoked to
explain the great Th/Os and Th/Ir ages. Os and Ir are in fact supposed
to
show only minor s-process contributions (Arlandini
et al. 1999). Again, we must
conclude that large systematic uncertainties still exist in the
theoretical
calculations of production ratios
.
![]() |
Figure 2:
Ages and spectroscopic age uncertainties for star CS 22892-052
determined from various chronometer pairs (symbols) assuming up to four
different production ratios. Filled symbols refer to the production
rations of Kratz
et al. (2007). The dashed line indicates the age of
the universe. Sneden
et al. (2003) give a radiochronometric age estimate
of |
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![]() |
Figure 3:
Same as Fig. 2
for star CS 31082-001. In addition, for this star U/Th ages are
available. Hill
et al. (2002) give a radiochronometric age estimate
of |
Open with DEXTER |
At this point one may ask what accuracy has been reached in spectroscopically measured abundance ratios necessary for radioactive dating - in particular for metal-poor stars -, and what are the perspectives for improving these accuracies in the future. Both questions are not easy to answer. Uncertainties given by individual authors are often on the optimistic side. We think that a comparison among different authors provides a more robust measure of the actually present - as opposed to published - uncertainties. Such a comparison might still be biased towards too small error bars since different authors often draw from the same sources for oscillator strengths, or may use the same observational material. Hence, the analyses are not independent. Sneden et al. (2009) performed such a comparison of the abundances which they obtained for the metal-poor r-process enriched halo giant CS 31082-001 with studies of Hill et al. (2002) and Honda et al. (2004). We repeat this comparison here for the obtained abundance ratios. In the comparison Honda-Sneden the list of common elements is La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Er, Tm, Yb, in Hill-Sneden La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Er, Tm, Hf. Abundance ratios among these elements can be taken as a rather good model for the Th/Hf pair since their lines emerge from the first, dominant ionization stage, and from levels of low excitation potential (Sneden et al. 2009; Lawler et al. 2009) - similar to the situation for the Th/Hf pair. Taking the abundance ratios from the individual studies ensures that correlation effects reducing the errors on abundance ratios are fully taken into account. For the rare earth elements often many lines are available for the determination of their abundance. This luxury is not afforded by Th or Hf so that we still have a likely bias towards too high accuracy when taking the rare earths as example.
From the lists of elements in common we formed all possible abundance ratios (except for an element with itself), and calculated the difference of the logarithmic ratios between the various studies. For the pair Hill-Sneden we find a dispersion among the differences of the logarithmic ratios of 0.14 dex, for Honda-Sneden 0.12 dex. In the case of Hill-Sneden, in particular the discrepant Hf abundances have to be noted. In the case of Honda-Sneden, a substantial part of the dispersion is driven by the elements Tm and Tb; leaving them out reduces the dispersion to 0.05 dex; however, this is not suggested by, e.g., excessive errors on their abundance as stated by the authors. We emphasize that at least in part the various groups use the same oscillator strengths. From this exercise we conclude that attributing an uncertainty of about 0.1 dex to the Th/Hf abundance ratio is perhaps conservative but not grossly over-estimating the actual value.
How to improve the accuracy? Sneden
et al. (2009) report a line-to-line scatter of
typically 0.07 dex in a given star when determining abundances
of a rare
earth element and applying the high-quality oscillator strength of
Lawler et al. (2009, and
references therein). A sub-optimal choice of the model
atmosphere introduces some scatter, but certainly errors in the
oscillator
strength are an important contributor. Since two abundances are
involved in
the determination of a ratio, an uncertainty of about 0.1 dex
would result
when an individual abundance is given to 0.07 dex accuracy. It
is often
argued that by averaging over abundances of N
individual lines one improves
the obtained accuracy by a factor .
However, this would only be the
case if the scatter among the oscillator strength is of random nature
which is
far from obvious. So, even more accurate oscillator strengths of the
involved
line transitions are desirable. This also relates to lines of blending
species
which one has to deal with. At this point an appropriate theoretical
description of the line formation process also comes into play which
should
account for departures from local thermodynamic equilibrium and
gas-dynamical
effects. All this is alleviated by having high signal-to-noise and high
resolution spectra at hand which ELTs will be able to provide. One
often has
to work with spectra where a line profile is sampled with only very few
points. Better sampling usually permits more accurate measurements. We
are
convinced that we are not stuck at the present level of accuracy to
which
abundance ratios used in radioactive dating can be determined but not
insignificant effort is necessary to make substantial progress.
4 The Solar age
As another application of the dating formulae we tried to deduce the solar age. We start by taking the Th/Hf pair as an example since we worked recently ourselves on this chronometer pair (Caffau et al. 2008) having all relevant spectroscopic information at hand. In the case of the 401.9 nm Th II / 409.3 nm Hf II line pair, our favorite spectral synthesis code gives CR / CX=0.13. For the Sun one has to consider two complications which are less severe for metal-poor stars:
- 1.
- the Th and Hf observed in the solar system are not the result of a single r-process event, but rather of the superposition of many events;
- 2.
- while Th is produced only through the r-process, Hf is in part produced by the s-process.




Turning now to spectroscopically measured abundances,
Table 1
lists the obtained ages and their uncertainties, again for different
chronometer pairs and assumptions about production ratios and r-process
fractions (see Tables A.1-A.3). The Th/Hf
age, 22.3 Gyr, comes out to be greater than the age of the
universe,
13.73 Gyr, as estimated from the fluctuations of the Cosmic
Microwave
Background (Spergel et al.
2007). By using the
value suggested by Lawler
et al. (2007), only 0.01 dex
higher than the abundance we use, the age comes out even a bit greater
by 0.5 Gyr.
Even if the solar abundance of Hf and Th were
produced in a number of distinct supernova events at different times,
the
resulting mean age as computed from Eq. (8) cannot be
greater than the time elapsed since the earliest supernova event, which
cannot
have occurred earlier than 13.73 Gyr ago. We have ignored the
destruction of Th by photons in stellar interiors. According to
del Peloso et al. (2005b)
the matter cycled through stars may have a reduction of
10-20% in Th. However, even when increasing the Solar Th abundance by
20%, one still derives an age of 18.2 Gyr from Eq. (2). We have to
conclude that either the assumed value of
,
or the r-process fraction of Hf is incorrect. Among
the other
chronometer pairs Th/Eu provides the closest match to the actual Solar
age,
while in general - like for Th/Hf - there it the tendency to grossly
overestimate the age. The photospheric solar abundance of europium
provided by Lawler
et al. (2007)
is exactly the same we are using.
Obviously, the complex chemical history of the Sun together
with uncertainties in the production ratios make it very difficult to
obtain a
reliable Solar age by radioactive dating.
In this context it is perhaps interesting to note that Sneden et al. (2009) find that the solar photospheric Hf abundance is deviating from the meteoritic value in contrast to all its preceeding (in atomic number) rare earth elements. Similar, the solar r-process abundance pattern corresponds to the one of r-process enriched iron-poor halo stars except for Hf which appears relatively depleted in the Sun. Of course, a higher r-process abundance of Hf in the Sun would even lead to a greater solar age aggravating the problem in the radioactive dating.
5 Conclusions and remarks
The accuracy of radioactive cosmo-chronometry is not going to be
extremely
high on a single star. A realistic estimate of what future
instrumentation
will be able to achieve does likely not allow one to hope for a better
than
2 Gyr precision, taking into account observational
uncertainties
only, leaving aside the systematics associated to production rates and
chemical enrichment models. Although high-mass n-capture
elements are
currently believed to be produced in a fairly well understood manner by
the
so-called ``main'' r-process (Kratz
et al. 2007; Farouqi et al. 2008),
the quantitative
determination of the production ratios
between the stable and the
unstable elements still bear significant uncertainties.
Since in photon noise dominated spectra the measurement error on the equivalent width decreases linearly with the observational S/N ratio, the attainable dating precision increases roughly linearly with the S/N ratio. This constitutes a major practical challenge to the application of the method, since many of the involved lines are in the blue part of the optical spectrum (e.g. for Th, U, and Hf), and the natural target of the analysis are cool stars.
If possible, the decaying element should be chosen such that
its mean lifetime
is similar
to the age one wants to measure. This applies also to the
``effective''
of a chronometric pair constituted by two decaying
species. From this perspective, the pairs U/Th, Th/stable, and U/stable
are
well suited to measure ages of old objects around 10 Gyr of
age. No other
naturally occurring isotope has a mean lifetime providing a similarly
close
match.
Potentially important systematic spectroscopy-related effects that we feel deserve further study to fully exploit the potentials of radioactive dating are effects related to departures from thermodynamic equilibrium and gas-dynamics on the line formation of the chronometric species. For instance, the application of a 3D radiation-hydrodynamical model atmosphere by Caffau et al. (2008) led to a downward revision of the photospheric Solar Th abundance by 0.1 dex. This corresponds to a decrease of Th ages by 4.7 Gyr.
Globular clusters provide fairly large samples of metal-poor, bright, coeval and chemically homogeneous stars. Simultaneous measurement of abundances in these stars allows us to reduce the observational, statistical uncertainties. This could permit to derive r-process production ratios empirically by calibration against the known ``photometric'' age. To our knowledge, this has never been attempted so far.
AcknowledgementsThe authors H.-G.L., P.B, and L.S. acknowledge financial support from EU contract MEXT-CT-2004-014265 (CIFIST).
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Appendix A: Tables of elemental abundances, production ratios, and solar r-process fractions
The following tables list the abundances, production ratios, and solar system r-process fractions that we compiled from the literature for the purpose of radioactive dating. In Table A.1, exclusively spectroscopically determined (as opposed to meteoritic) abundances are given for the Sun. In Table A.2 of production ratios, the values for Kratz et al. (2007) refer to the ones listed as ``Fe-Seed fit2'' in Table 2 of their paper. Kratz and collaborators report three different estimates of the various production ratios.
Table A.1: r-process elemental abundances in metal-poor stars and the Sun for radioactive dating.
Table A.2: r-process production ratios.
Table A.3:
Solar r-process fractions
for various elements X.
All Tables
Table 1:
Ages and age uncertainties
of metal-poor stars and the
Sun obtained for different chronometer
pairs X/Y.
Table A.1: r-process elemental abundances in metal-poor stars and the Sun for radioactive dating.
Table A.2: r-process production ratios.
Table A.3:
Solar r-process fractions
for various elements X.
All Figures
![]() |
Figure 1:
The function |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Ages and spectroscopic age uncertainties for star CS 22892-052
determined from various chronometer pairs (symbols) assuming up to four
different production ratios. Filled symbols refer to the production
rations of Kratz
et al. (2007). The dashed line indicates the age of
the universe. Sneden
et al. (2003) give a radiochronometric age estimate
of |
Open with DEXTER | |
In the text |
![]() |
Figure 3:
Same as Fig. 2
for star CS 31082-001. In addition, for this star U/Th ages are
available. Hill
et al. (2002) give a radiochronometric age estimate
of |
Open with DEXTER | |
In the text |
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