Stellar Mixing and the Primordial Lithium Abundance
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Stellar Mixing and the Primordial Lithium Abundance M.H.Pinsonneault 1,G.Steigman 1,2,T.P.Walker 1,2,&V.K.Narayanan 3January 112001ABSTRACT We compare the properties of recent samples of the lithium abundances in halo stars to one another and to the predictions of theoretical models including rotational mixing,and we examine the data for trends with metal abundance.We apply two statistical tests to the data:a KS test sensitive to the behavior around the sample median,and Monte Carlo tests of the probability to draw the observed number of outliers from the theoretical distributions.We ?nd from a KS test that in the absence of any correction for chemical evolution,the Ryan,Norris,&Beers (1999)(hereafter RNB)sample is fully consistent with mild rotational mixing induced depletion and,therefore,with an initial lithium abun-dance higher than the observed value.Tests for outliers depend sensitively on the threshold for de?ning their presence,but we ?nd a 10??45%probability that the RNB sample is drawn from the rotationally mixed models with a 0.2dex me-dian depletion with lower probabilities corresponding to higher depletion factors.When chemical evolution trends (Li/H versus Fe/H)are included in our analy-sis we ?nd that the dispersion in the RNB sample is not explained by chemical evolution;the inferred bounds on lithium depletion from rotational mixing are
similar to those derived from models without chemical evolution.Finally,we ex-plore the di?erences between the RNB sample and other halo star data sets.We ?nd that di?erences in the equivalent width measurements are primarily respon-sible for di?erent observational conclusions concerning the lithium dispersion in halo stars.Implications for cosmology are discussed.We ?nd that the standard
Big Bang Nucleosynthesis predicted lithium abundance which corresponds to the
deuterium abundance inferred from observations of high-redshift,low-metallicity
QSO absorbers requires halo star lithium depletion in an amount consistent with
that from our models of rotational mixing,but inconsistent with no depletion.
Subject headings:stars:abundances;cosmology:cosmological parameters;stars:
rotation
1.Introduction
The primordial abundance of the light element lithium provides a crucial test of Big Bang nucleosynthesis(BBN);it is also an important diagnostic of standard and nonstan-dard stellar evolution theory.The detection of7Li in halo stars by Spite&Spite(1982) opened up the prospect of the direct detection of the primordial lithium abundance.There have been a number of subsequent observational e?orts which have produced a detailed pic-ture of the distribution of halo star lithium abundances(Spite&Spite(1993);Thorburn (1994)(hereafter T94);Bonifacio&Molaro(1997);RNB;see also Ryan et al.(1996)). Primordial7Li,as one of the four light nuclides produced in measurable abundance in stan-dard BBN(the others being D,3He,and4He??see Olive,Steigman,&Walker(2000) for a review),provides a crucial consistency check in that all four nuclides are determined by the one free parameter of standard BBN??the baryon-to-photon ratio,η.Currently, the primordial deuterium abundance provides the best estimate ofη.However,the BBN-predicted primordial lithium abundance which is consistent with the observationally inferred primordial deuterium abundance(and thus with our best estimate ofη)is actually much larger than the lithium abundance observed in halo stars.We show that theoretical models which include rotational mixing(and are required by the observed dispersion of halo lithium abundances)predict a primordial lithium abundance which is consistent,in the context of standard BBN,with the observed primordial deuterium abundance.
1.1.Stellar Models Compared with Earlier Data Sets
The interpretation of the halo star data requires knowledge of the stellar evolution e?ects which have in?uenced the surface abundance during the lifetime of the stars.In“classical”
(i.e.,nonrotating)stellar models lithium is destroyed on the main sequence only in the presence of a deep surface convection zone;some pre-main sequence depletion will occur
for a wider range of masses.Such models predict only small amounts of lithium depletion for the hottest subdwarfs(e?ective temperature greater than about5800K)and for their Population I analogs(e.g.,Deliyannis,Demarque,&Kawaler(1990)).In the Population I case classical models make detailed predictions about lithium depletion which can be tested using data from open clusters with a range of ages.The open cluster data is in strong contradiction with the predictions of classical models.In particular,there is observational evidence for a dispersion in lithium abundance at?xed mass,composition,and age,and also for lithium depletion on the main sequence in stars with surface convection zones too shallow to burn lithium in the classical models(e.g.Balachandran(1995);Pinsonneault(1997); Jones,Fischer,&Soderblom(1999)).The rate of main sequence depletion is observed to decrease with age,and there are also strong mass-dependent depletion e?ects,none of which are predicted by the classical stellar models.
A number of physical mechanisms neglected in classical stellar models have been sug-gested as possible causes for the discrepancies.Rotational mixing is one attractive explana-tion since a range in initial rotation rates will produce a range in rotational mixing rates and the rate of rotational mixing would decrease with age as low mass stars lose angular momen-tum.Unfortunately,models of Population II stars cannot be subjected to the same stringent level of tests that can be performed for open cluster stars.The major unique signature of rotational mixing in the Population II context is therefore the presence of a dispersion in lithium abundance at?xed mass and composition.As a result,much of the theoretical and observational work on the subject has therefore focused on the existence and magnitude of dispersion in the Population II lithium abundances.
In a previous paper(Pinsonneault et al.(1999),hereafter PWSN)we computed the distribution of7Li depletion factors expected from stellar models including rotational mixing. The distributon of depletion factors was compared with the largest uniform data set available, that of T94.We concluded that a combination of the observed dispersion in abundances,the relative depletion of the isotopes7Li and6Li,and the existence of a small population of highly depleted stars all argued in favor of the stellar depletion of lithium and we placed bounds of0.2??0.4dex on the7Li depletion factor.In this paper we compare our theoretical calculations with the newer halo lithium data set of RNB.
The principal properties of lithium depletion in stellar models which include rotation can be summarized as follows.Rotation can induce mixing in the radiative interiors of stars leading to surface lithium depletion during the main sequence phase of evolution.This depletion due to rotational mixing is in addition to surface lithium depletion during the pre-main sequence and(in the case of cool stars)main sequence evolution.The degree of rotational mixing depends on the angular momentum content and its evolution so that a
range of pre-main sequence rotation rates will produce a range of lithium depletion factors, in the sense that rapid rotators experience more mixing and lithium depletion than do slow rotators.There is compelling evidence from the Population I data for main sequence lithium depletion as well as for a dispersion in lithium abundance at?xed mass,composition,and age;rotational mixing naturally explains this pattern.PWSN found that halo star models experience systematically less lithium depletion than do solar abundance models for the same sets of initial conditions.
The distribution of lithium depletion factors depends on the distribution of initial con-ditions,which can be inferred for young Population I clusters.The distribution of pre-main sequence rotation rates needed to reproduce the rotation data in the Pleiades cluster pro-duced a degree of dispersion which is correlated with the absolute amount of7Li depletion. Since the majority of young stars have similar rotation rates,the majority of stars will ex-perience similar7Li depletions.There is,however,a subpopulation of rapid rotators that are predicted to experience higher7Li https://www.sodocs.net/doc/1a16075989.html,parison with the T94data set led to a range of0.2??0.4dex in the inferred stellar depletion(PWSN).When combined with an observed“Spite plateau”7Li abundance of2.25±0.10(on the logarithmic scale where H=12.0),this yielded a primordial7Li abundance in the range of2.35??2.75.We em-phasize that the PWSN models have the following overall properties:in contrast to a simple gaussian distribution of abundances there is a distribution with a core whose dispersion is dominated by observational errors,along with a subpopulation(of order1/5of the sample) with moderately higher depletion factors,and a smaller population of(order2-3%of the sample)with large depletion factors.These features will prove important in our comparison with newer halo star data of RNB.
1.2.New Results from RNB
The PWSN conclusions have recently been challenged by RNB using data from a high precision study of lithium abundances in a smaller,albeit still signi?cant,sample of halo stars.They obtained both a lower absolute observed abundance(2.11)and a signi?cantly reduced error estimate and dispersion.They attributed the residual dispersion to chemical evolution(e.g.the observed spread in Li/H in their view is caused by di?erences in post-BBN lithium production correlated with the range in Fe/H).They argue that their data set requires stellar depletion be minimal.In recent papers(Ryan et al.(2000),Suzuki,Yoshii,& Beers2000)the RNB results have been used to argue that the primordial lithium abundance is below their observed value in very metal-poor stars as a result of galactic production.In this paper we compare our models with this new data set and we also compare the RNB
data set with other studies.We begin by comparing the data set of RNB with the theoretical distributions of PWSN in section2.We analyze the dispersion and chemical evolution trends in section3and compare the RNB and T94data sets in section4.Our conclusions concerning the primordial abundance of lithium and its consequences for cosmology are found in section 5.
https://www.sodocs.net/doc/1a16075989.html,parison of the Models with the Data
The RNB sample does have a dispersion in excess of their observational errors.RNB attribute this excess dispersion to chemical evolution.We will begin by comparing the RNB data to theory without any chemical evolution detrending;we consider both the reality of any trend with metallicity and its impact on the inferred lithium abundance in section3.We note here that none of the overall conclusions of the comparison between data and theory in this section are dramatically modi?ed by the treatment of chemical evolution e?ects(see section3.)Furthermore,an analysis of the models without metallicity detrending provides the least model-dependent constraint on the degree of rotational mixing.
Di?erences in stellar rotation rates will produce di?erences in the degree of rotational mixing,so excess dispersion can be a signature of stellar depletion.However,the distribution of stellar rotation rates is needed in order to predict the distribution of stellar lithium depletion factors.Stellar models with rotation must also account for angular momentum loss from a magnetic wind and internal angular momentum transport.Finally,the degree of mixing for a given angular momentum distribution must be speci?ed(see PWSN for a more detailed description.)
As discussed in PWSN,rotation data in young open cluster stars is our best current guide to the initial conditions that might be applicable to halo stars.The majority of young stars are slow rotators with similar rotation rates;these stars will have almost uniform deple-tion and very little internal scatter.About15%of young stars are rapid rotators,including a subpopulation(about3%)of very rapid rotators.This will produce a tail of overdepleted stars in the distribution.There are unavoidable observational selection e?ects which may in?uence the inferred distribution of rotation velocities.For example,very slow initial rota-tors would only have upper limits to their rotation velocity,so it is di?cult to estimate how many stars should be underdepleted compared to the median.There are occasional claims of pre-MS stars with very long periods,and this might explain the occasional halo star above the lithium plateau.At the other end,the rapid rotator tail is subject to Poisson noise??the fastest spinner in the Pleiades is at140km/s and the second fastest is at90km/s.So the very far tail of the underdepleted stars is di?cult to pin down.However,the behavior
of the peak of the distribution is not sensitive to these details.This provides justi?cation for our including the one upper limit lithium abundance in the RNB sample and considering those outliers below,but not above,the median in our tests of the models.
We can empirically constrain the angular momentum loss and transport properties by comparing di?erent classes of theoretical models to stellar observations as a function of mass and age.Angular momentum transport and mixing by hydrodynamic mechanisms are included in the models.We calibrate the mixing by requiring that a solar model reproduce the solar lithium depletion at the age and rotation rate of the Sun.However,we have no direct information on the solar initial conditions;because angular momentum loss scales asω3,stars with a wide range of initial rotation rates end up with similar rotation rates at old ages.In PWSN,we considered three solar calibrations(s0,s0.3,and s1)which correspond to three di?erent overall normalizations for the stellar lithium depletion.The s0 case assumes that the Sun was initially a rapid rotator,so the typical star will experience much less depletion than the Sun;the s0.3and s1cases correspond to assuming the Sun is more typical and the overall expected stellar depletion is therefore larger.
The Pinsonneault,Deliyannis,&Demarque(1992)depletion factor of10from rota-tional mixing came from the assumption that the Sun was a typical star;furthermore,these early models did not include a saturation of angular momentum loss for rapidly rotating stars.The current generation of models is in signi?cantly better agreement with more recent measurements of stellar rotation rates,which both permits us to infer the distribution of rotation rates and to rule out depletion factors as large as the1992values.
There will be two principal di?erences between rotationally mixed and standard models that can be directly tested with the halo star lithium data.The internal range in rotation among slow rotators will produce an increase in the dispersion around the sample median relative to the observational errors;and the rapid rotators will be overdepleted relative to the median.We therefore apply two statistical tests to the data.We compare the cumulative distribution of stars to theoretical distributions anchored at the median abundance of the sample using a KS test.This test allows us to measure the constraints on stellar depletion from the tightness of the bulk of the halo lithium plateau stars.
We also applied both a simple analytical model and Monte Carlo simulations to test the probability of drawing the observed number of outliers from the theoretical simulations.For a given distance below the median there is a probability that any given star in the theoretical distribution will lie at or below that abundance(P0).For a sample of size N,the probability that there will be a given number I of stars below such a threshold is S I P I0(1?P0)N?I where S I is the number of states capable of producing a given number of outliers.For I=0,1, S I=1,N respectively;it is straightforward if tedious to compute the number of accessible
states for more outliers.We used Monte Carlo simulations to check for the cases with larger numbers of outliers.
We also compare to a Gaussian distribution of errors.This permits us to test for the possibility that the excess dispersion arises from a global underestimate of the observational errors and to quantify the relative agreement of models with and without stellar depletion.
https://www.sodocs.net/doc/1a16075989.html,parison with the Cumulative Distribution
We convolved the theoretical distributions for the s0,s0.3,and s1cases of PWSN de-scribed above with gaussian observational errors of0.035dex(see Section4for our determi-nation of the observational errors).In Figure1we compare the cumulative RNB distribution with these three models and a Gaussian distribution of errors.The s0,s0.3,and s1models have median depletion factors of0.18dex,0.32dex,and0.50dex respectively;on the RNB abundance scale these would correspond to initial abundances of2.29,2.43,and2.56respec-tively.Despite the very di?erent depletion factors,all of the models have similar properties in the core of the distribution.Only the s1case,with a high median stellar depletion of0.50 dex,predicts a core broader than the observed distribution.Gaussian errors alone cannot reproduce both the tightness of the core and the presence of outliers in the sample.
KS tests applied to this distribution indicate that there is,respectively,a60%,a25%, and a5%chance that the s0,s0.3,and s1cases could be drawn from the same distribution as the data;by comparison the Gaussian has an86%chance of being drawn from the same distribution as the data.We conclude from this test,which is primarily sensitive to the behavior of a sample around the median,that only high depletion factors are problematic??although even the highest depletion case is only excluded at the95%level.This con?dence level is too low to absolutely rule out a model.There is thus no contradiction between mild stellar depletion from rotational mixing and the presence of a core of halo lithium abundances with small internal scatter.This data suggests that the internal dispersion in the core is primarily caused by observational error,and furthermore that an underestimate of the observational errors is not responsible for the excess scatter in the data.At the same time,it also provides no support for depletion factors at or above the0.5dex level.
2.2.Number of Outliers
Inspection of Figure1reveals that the theoretical distributions predict more overde-pleted stars than are present in the data set,but that there are more overdepleted stars
in the sample than are predicted by the observational errors(compare the solid and dotted curves).Because of the small number of stars(23)in the sample,and the even smaller number of outliers expected in the sample(3??7),we believe that only tentative claims can be made about the consistency(or lack thereof)of modest depletion factors with the data.The basic issue is simply that the expected number of outliers in the rotational mixing models is small for a sample of23stars,which makes the conclusions subject to Poisson noise.
As an illustration,consider the di?erent distributions in Figure1.The tail of the observed distribution up to an abundance of2.0corresponds to three overdepleted stars; it is clearly inconsistent with the expectation from observational errors,since abundances this low are formally threeσor more below the median and therefore very unlikely in a sample of23stars.The highest depletion case predicts more stars more than0.1dex below the median(7)than are observed(3).However,the sample is so small that the speci?c statistical conclusions depend sensitively on where the threshold for de?ning an outlier is de?ned.If the threshold is de?ned at2.01(just above two of the three overdepleted stars), then the expected fraction of outliers relative to the data is minimized and there is a45% chance of drawing the observed number of stars relative to the s0case and less than a0.1% probability of seeing as many as three outliers from observational errors alone.
However,there is a gap in the sample between abundances of2.00and2.06;if an outlier is de?ned as being at or below2.05the expected outlier fraction is increased and the observed outlier fraction is the same.In this case there is a10.6%chance of drawing the data from the minimally depleted s0distribution and an11.4%chance of seeing as many as three outliers from observational errors.Similar?uctuations arise from excluding the one upper limit from the sample or clipping the tail of the theoretical distribution.We therefore consider a range of probabilities from the most stringent(counting all stars more than0.06dex below the median as outliers)to the least stringent(counting all stars more than0.10dex below the median as outliers.)
The numbers in parenthesis after the listed fractional probabilities in the second and third columns of Table1are the expected number of outliers if we set the threshold for de?ning one at less than2.06or less than2.01respectively.The actual number of outliers is three below2.01or below2.06,e.g.,there are no stars between2.00and2.06.The probabilities for the Gaussian are for having three or more outliers;the probabilities for the other three cases are for having three or fewer outliers.Because of sparse sampling there is a range of possibilities for de?ning what an outlier is.The closer the cut is to the median, the larger the number of expected outliers;this favors the no-depletion case because there are more outliers than expected,but disfavors the stellar depletion case because there are
fewer outliers than expected.
If we exclude the one upper limit(G186-26),the minimum/maximum probabilities for the s0case drop to5.5%and24.8%respectively for two outliers out of22stars.Ryan et al. (2001)have argued that the rare ultra-lithium depleted stars are binary merger products, and that they should therefore be excluded from samples of this type.In support of this they note that there is a large di?erence in abundance between the ultra-depleted stars(of order5%)and others and that the fraction of overdepleted stars is very high in intermediate metal abundance stars which are hot enough to be plausible blue straggler candidates.
We?rst note that Ryan et al.(2001)does not establish a causal link between high lithium depletion and binary merger products;in fact,the authors argue that excess lithium depletion may be the sole indicator of such processes.The fraction of highly depleted stars in some clusters,such as M67(Jones,Fischer,&Soderblom(1999))is signi?cantly higher than the norm,which could indicate at minimum that there is more than one cause for strong lithium depletion.Excluding stars that do not?t an expected pattern also amounts to an e?ective prior on the sample statistics.If highly depleted stars are a priori excluded from lithium samples then the highly depleted stars predicted by theoretical models should also be removed when doing statistical comparisons.If we remove the observed upper limit and the upper5%of depletion factors from the theoretical models on the grounds that they are rejected from samples of lithium abundances,we recover the same(or higher)probabilities as we infer from including the one upper limit in the sample.Therefore,in this and subsequent tests we retain the entire sample for statistical comparisons.
In constrast to the behavior around the sample median,the number of outliers sets stronger constraints on stellar depletion.The highest depletion case of0.5dex is ruled out at the95%con?dence level even if we use the most generous de?nition of what constitutes an outlier;this is a considerably stronger test than a similar con?dence level for a KS test. The Gaussian error model has severe di?culty reproducing the observed number of outliers; it is excluded at the90%con?dence limit0.05dex below the median and at higher than a 99.9%con?dence level0.10dex below the median.Formally,a model with0.13dex depletion would provide the best?t to the outlier fraction.The0.32dex depletion case is not formally excluded,but it is certainly disfavored by the present data set.
We can set some rough bounds on stellar depletion based on the RNB data sample without considering the e?ects of chemical evolution(discussed in Section3)or possible systematic di?erences in equivalent width measurements(see Section4.)Stellar depletion at the0.1dex level is fully consistent with the data.Models with depletion as high as0.5dex are less than5%probable,while models with no depletion are less than10%probable.PWSN compared the same models with the full T94data set and concluded that stellar depletion
at the0.2dex level provided the best?t to the dispersion in the T94data set;a range of 0.2to0.4dex depletion was the result of several di?erent diagnostics of stellar depletion including the presence of highly depleted stars and6Li to7Li ratio measurements and limits. The base RNB data set provides a lower central value for depletion;but because of the small sample size the bounds on depletion are actually widened relative to the conclusions of PWSN.In the next sections we consider other e?ects,and will return to our?nal estimate of the primordial lithium abundance in section5.
3.Trends with Metal Abundance
The dispersion in the RNB sample exceeds their quoted observational errors;as we have shown above it is consistent with the theoretical predictions of mild rotational mixing. However,RNB concluded that the excess dispersion in their sample could be explained instead by post-BBN galactic production of lithium.As evidence for this they performed ?ts of lithium versus iron adopting for the functional form a?t which is a power-law in Li/H versus Fe/H:log(Li/H)=log(Li/H)P+a[Fe/H],where[Fe/H]≡log(Fe/Fe⊙).
Although this form may provide a good?t to the data over a limited range in metallicity, it certainly cannot describe the evolution of an element whose BBN abundance is expected to provide the dominant contribution to its halo abundance.To account for a signi?cant BBN component along with a chemical evolution component that may scale linearly with the iron abundance(see for example Ryan et al.(2000)),the?tting function should be of the form Li/H=(Li/H)P+b(Fe/Fe⊙).Since post-BBN,early galactic production of lithium may be dominated by cosmic ray nucleosynthesis which depends more on the oxygen than the iron abundance,Ryan et al.(2000)also considered the consequences of an increasing oxygen abundance at low iron abundance.In this case a linear?t to lithium as a function of oxygen would take the form Li/H=(Li/H)P+b(Fe/Fe⊙)0.7.They found signi?cant slopes ranging from4.0×10?9to1.8×10?8(in the linear iron??linear lithium plane) and ranging from0.9×10?9to3.4×10?9(in the linear oxygen??linear lithium plane) on the assumption that the controversial claims of very high oxygen abundance at low iron abundance are correct(Israelian,Garcia-Lopez,&Rebolo(1998);Boesgaard et al.(1999); but see also Fulbright&Kraft(1999),King(2000)).Although we obtain somewhat smaller slopes,we will show that the most important feature of these chemical evolution?ts is that they do not explain the outliers seen in the RNB sample.Therefore,in contrast to RNB, we?nd that chemical evolution cannot account for the excess dispersion observed in their sample.
There are several issues which e?ect the quantitative?ts for the possible early(low-
metallicity)evolution of lithium.For example,the?ts depend on the adopted stellar metal-licities and RNB included two sets of metallicity estimates.A literature value was taken from Ryan&Norris(1991),Ryan,Norris,&Bessel(1991),Carney et al.(1994),and(for one star)Beers,Preston,&Shectman(1992).There was also a1angstrom resolution estimate directly obtained by RNB for21/22detections in their sample.Since no error estimates are quoted in the paper,we estimated them in two ways.The rms di?erence between the two sets is0.14dex,consistent with a1σerror of0.1dex in each.This is also consistent with the error estimates in the primary sources used by RNB for the“literature”values.Furthermore, there is a zero-point di?erence of0.13dex between the literature and RNB metallicities,in the sense that the RNB values are higher.This di?erent metallicity zero-point contributes to the range in the inferred chemical evolution slopes,in the sense that the slope inferred from the RNB metallicities is smaller than that obtained with the literature metallicities. Because the corrections to the lithium abundances are small,only in the literature case does the error in the metallicity have an impact on the overall dispersion(raisingσfrom0.035to 0.040in the most extreme case.)
In addition,the RNB literature metallicities have an embedded e?ect that produces a signi?cant component of the higher slope.Two of the sources??Ryan&Norris(1991), Ryan,Norris,&Bessel(1991)??are systematically0.15dex lower than Carney et al. (1994)because of a di?erence in the assumed solar iron abundance.The Ryan,Norris,& Bessel(1991)abundances were corrected to the Carney et al.(1994)scale,but the Ryan &Norris(1991)values were not;RNB also did not use RN91or Carney when there was an abundance from Ryan,Norris,&Bessel(1991).To test for the importance of this e?ect we used the same primary sources,but corrected Ryan&Norris(1991)to the Carney et al. (1994)scale.We then averaged multiple measurements weighted by their respective errors. This reduces the rms scatter compared with the RNB1A metallicities by25percent,and the slope also drops by25percent.The direct1σerror in[Fe/H]is0.08dex.We therefore conclude that half of the di?erence between the literature and RNB abundances is caused by the combination of data from di?erent sources in the RNB literature values and the other half is the metallicity zero point.We use the published literature RNB data for comparison with other papers that have used this data;we believe that the homogeneous RNB metallicities are a better choice for chemical evolution studies.
In Table2we show the data we used.The abundance errors were estimated by adding in quadrature the T e?error,the RNB slope of0.065dex per100K,and the RNB equivalent width error in the linear curve of growth approximation.We obtain an average sample error of0.036dex rather than the RNB value of0.032dex;we have not been able to trace the origin of the latter number in the RNB paper.The T94abundances have been converted to the RNB temperature scale using the temperature correction above;the T94errors were
estimated from the T94equivalent width errors and the temperature errors as described above.The average T94error is0.06dex;we defer a discussion of the T94data to section4.
We considered two sets of metal abundances and two?tting functions in T e?(linear and power law),for four basic cases.As already noted by RNB,much of the slope comes from three outliers in the sample;we therefore repeated the analysis for all four sets with the same outliers excluded as discussed in RNB.In Table3we present the eight sets of results.The cases are identi?ed in column1;the?rst four include outliers and the last four do not.The cases starting with L use the RNB[Fe/H]metal abundances which yield a low slope;the cases starting with H use the literature[Fe/H]values which yield a high slope.The cases ending with L are linear?ts and the cases ending with P are power law?ts.The zero-point and slope of the di?erent?ts are in columns2and3.The median abundance corrected to zero metal abundance and both the predicted and actual residual dispersion are given in columns4to6.The di?erent cases are illustrated in Figures2and3.
There are two important conclusions to be drawn from this exercise.First,the chemical evolution slopes are sensitive to all of the assumptions in the models,with a wide range of slopes possible.Second,detrending the data in the linear Li versus Fe plane does not bring the outliers onto the mean trend.In all cases the formal dispersion of the detrended samples are larger than the estimated errors.This can be traced directly to the presence of outliers whose lithium abundance di?ers signi?cantly from the sample mean.Intuitively this can be easily understood;because the absolute metal abundances of the stars are small,there is little room for a signi?cant chemical evolution correction.There are three stars noted as outliers in the RNB chemical evolution analysis:CD-2417504([Li]=1.97±0.033,4.2σbelow the mean);BD+92190([Li]=2.0±0.042,2.6σbelow the mean);and CD-711234 ([Li]=2.20±0.025,3.6σabove the mean).In the linear?t to the literature iron abundances these three stars are respectively2.4σbelow,2.3σbelow,and2.0σabove the mean;for the linear?t to the RNB iron abundances the same stars are respectively3.3σbelow,2.3σbelow,and3.8σabove the mean.There is also the upper limit in G186-26,making a total of 4/23outliers regardless of the presence or absence of chemical evolution detrending.Similar results apply to the power-law?ts.
We have not performed chemical evolution?ts for our rotationally mixed models be-cause it is not a well-posed problem for such a small sample of lithium abundances;a metallicity-dependent distribution of stellar depletion factors needs to be convolved with a mean chemical evolution trend.From PWSN,we can anticipate that the contribution of a range of metallicities to the dispersion will be small and di?cult to detect in a sample of this size.
https://www.sodocs.net/doc/1a16075989.html,parison of Theory and Observation Including Chemical Evolution
Detrending
Table3presents the results of KS and outlier test comparisons of the models and data under di?erent chemical evolution detrending scenarios.We have used the same theoretical models as in section2.As noted in Table3,the additional observational errors from the uncertainty in lithium production does not signi?cantly impact the overall observed error. Because the various?ts yield similar conclusions,in Figure4we show only the most probable of these cases.The observational data in in Figure4is the cumulative distribution of[Li] from the linear?t to the RNB metal abundances,corrected to zero metal abundance.We compare this data set to a gaussian withσ=0.04dex and the same three theoretical distributions as in Figure1.The qualitative trends are similar to those obtained with the base RNB data.
The?ve cases considered are no chemical evolution;low slope,linear Li-Fe(LL),low slope,power law Li-Fe(LP),high slope linear(HL),high slope power law(HP).The?rst three columns are the probabilities of drawing the data from the theoretical s0,s0.3,and s1distributions.The no evolution case is evaluated at0.1dex below the median;the other cases are smoother and are evaluated in0.01dex increments between0.05and0.1dex below the median and averaged.The second set of three columns are the KS test probabilities for the same cases and theoretical distributions.
In Table5,we give both the zero points and inferred primordial abundances for the di?erent cases on the RNB abundance scale;we argue elsewhere that these should be adjusted up by0.1dex because of systematic model atmosphere/temperature scale e?ects.
3.2.Chemical Evolution Implications for6Li
If,indeed,the abundance of lithium is evolving at very low-metallicity as RNB suggest, the most likely source of post-BBN lithium is from cosmic ray nucleosynthesis(Reeves, Fowler,&Hoyle(1970)).One consequence of CRN is the concommitant production of6Li along with7Li resulting in comparable amounts of post-BBN production of both isotopes.At very low-metallicity the lithium isotope production is dominated byα?αfusion(Steigman and Walker1992)leading to a7/6production ratio of R76≈1.6(Kneller,Phillips,and Walker2000).At higher metallicity this ratio decreases slightly to R76≈1.5(Steigman and Walker1992;Kneller,Phillips,and Walker2000).As a result,the6Li/7Li ratio provides a means to test the RNB hypothesis that the lithium abundance is increasing at a noticeable rate in the early Galaxy at very low metallicity([Fe/H] ?2).If the observed lithium
abundances(without allowance for depletion by rotational mixing)are?t to a metallicity relation of the form Li/H=a+bx,where a≡(Li/H)P and x is either Fe/Fe⊙or(Fe/Fe⊙)0.7, the predicted7/6ratio is
(1+R76)a
7Li/6Li=R
76+
Spite&Spite(1993)sample(Spite et al.(1996),Spite&Spite(1982),Spite,Maillard, &Spite(1984);hereafter SS).Because the Ryan et al.(1996)sample is dominated by the large number of stars from the T94sample,there are really only two independent,primary samples which may be compared with RNB:T94and the SS sample.
https://www.sodocs.net/doc/1a16075989.html,parison with Other Data Sets
https://www.sodocs.net/doc/1a16075989.html,parison with T94:The Origin of Di?erences in Zero-point and Dispersion
The conclusions drawn by RNB and T94are markedly di?erent despite the signi?cant overlap in the two samples and their similar goals and design.We therefore begin by exam-ining the ingredients that could be responsible for this di?erence,namely1)the statistics of the subset of the T94data set studied by RNB,relative to the statistics of the entire T94 data set;2)the choice of e?ective temperature scale;3)the equivalent width measurements and;4)the model atmospheres used to relate equivalent width and e?ective temperature to abundance.
The raw dispersion measured by RNB for their sample of22stars(0.052dex)is very similar to the raw dispersion for the subset of their18stars in common with T94(0.054 dex),suggesting that the stars not in common do not strongly in?uence the overall result. The raw dispersion for the full T94sample was0.13dex,similar to the dispersion of0.12 dex that would be inferred for the subset of18stars from the T94data.Therefore,the RNB data set appears to be a fair subsample of the T94data set.This is not surprising since both were chosen using similar kinematic,metal abundance,and e?ective temperature criteria.However,the average abundances for the stars in common derived by RNB and by T94di?er signi?cantly,2.11and2.29respectively.
In subsequent steps we examined the impact of changes in the temperature scale and equivalent widths.The T e?scale chosen by RNB is di?erent(and on average cooler)than that used by T94.To estimate the importance of this e?ect,we compared the tempera-tures used by RNB and T94.We used the RNB slope of0.065dex per100K to infer the lithium abundances that T94would have obtained using the RNB temperature scale.If the di?erent temperatures adopted by T94and RNB were responsible for the di?erent conclu-sions about the sample dispersion we would expect a large decrease in the sample dispersion by performing this operation while retaining the T94equivalent widths and model atmo-spheres.Adopting the RNB temperatures reduces the average abundance inferred using the T94equivalent widths and atmosphere model only from2.29to2.25,while actually slightly increasing the dispersion that would have been inferred from the T94data relative to the
T94T e?scale.The abundances that would have been inferred from the T94equivalent widths and model atmospheres with the RNB T e?scale are given in Table2(see section 2).We conclude that while the choice of temperature scale does in?uence the abundance zero-point,the di?erent temperature scales do not explain the di?erence in the dispersion of the samples.We illustrate this in Figure6,where the RNB abundances are compared with the T94abundances for stars in common shifted to the same T e?scale.The intrinsic scatter is clearly larger for the T94equivalent widths,even accounting for the larger formal equivalent width error bars.
We also derived the abundances that T94would have obtained had both the lithium equivalent widths and temperatures of RNB been used instead of the EW and T e?as adopted by T94.The sole remaining di?erence after this has been done is the choice of model atmospheres relating equivalent width,temperature,and abundance.For this test we used the linear curve of growth approximation;e.g.the corrected[Li/H]=[Li/H](T94)+log(EW RNB/EW T94).Changing the equivalent widths(along with T e?)leads to a large decrease in the dispersion,from0.13dex to0.07dex;furthermore,the average inferred abundance decreases to2.22.The average di?erence between the equivalent width measurements of RNB and T94is1.9mA(in the sense that T94is systematically higher),so there is both a zero-point shift and a di?erence in the range of equivalents widths at?xed T e?in the RNB sample relative to the T94sample.
We attribute the remaining zero-point shift to one of two e?ects.The linear curve of growth assumption that we have employed could introduce some errors;T94and RNB also used di?erent model atmospheres to relate abundance to equivalent width.Ryan et al. (2000)estimate the systematic di?erences arising from the di?erent model atmospheres to be at the～0.08dex level,which can account for all but a small(0.03dex)di?erence in the mean abundance.We therefore conclude that the reason for the signi?cantly di?erent RNB dispersion estimates from those of T94are due to di?erences in the underlying basic equivalent width data and not by the choice of T e?,the sample properties,or the model at-mospheres.In contrast,the di?erence in the lithium abundance zero-point can be attributed to a combination of a di?erent T e?scale,systematically lower RNB equivalent widths relative to T94,and the choice of di?erent model atmospheres.This leads to a large overall di?erence between the mean abundances derived for stars in common,corresponding to a change in the inferred primordial lithium abundance comparable to the lower end of the range of stellar depletion presented in PWSN.
There is an average zero-point o?set of2.5mA in the RNB sample relative to the SS data set;a similar e?ect compared was discussed in Ryan et al.(1996).The overall morphology is similar to that between T94and RNB:2out of10points di?er by more than2σeven
when the zero-point o?set is taken into account.For completeness,we note that there is also a zero-point o?set of0.8mA relative to the Ryan et al.(1996)analysis;as mentioned above,because this sample is heavily weighted by the T94sample we did not perform a separate comparison of the Ryan et al.(1996)and RNB samples.We therefore conclude that zero-point di?erences in equivalent width measurements appear to be signi?cant,and by themselves they contribute an uncertainty of order10%to the absolute abundances.
4.2.Interpretation of the Di?erences
Even before reducing the dispersion by appealing to chemical evolution trends,the RNB sample has a smaller dispersion than does the T94sample.RNB attributed the di?erences between their data and that of T94to an underestimate of the errors in the T94data.In particular,T94did not correct for scattered light and sky subtraction.RNB note that this could increase the formal errors of individual data points in the T94sample by a factor up to1.7.In this case,one would not expect a gaussian distribution of the di?erences in equivalent widths,since the T94stars with the largest relative errors from these e?ects would be a?ected more than those where the quoted T94error estimates are accurate.There is an independent test of this hypothesis:we can compare the results of SS with those of T94.If the most discrepant points are due to larger than expected errors in the T94measurements, then there is no reason to expect the SS sample to have encountered the same problems.
In Figure7we compare equivalent width measurements for stars from di?erent sources (RNB,T94,SS).The three left-hand panels compare the eight stars with measurements from all three sources;the right panels compare RNB with the stars in common with the T94and SS data sets,respectively.Although the overlap among the samples is small(eight stars),we see no direct evidence that the T94data is in con?ict with the two other data sets;similarly,Ryan et al.(1996)found a good correlation between the SS data and the T94data once a zero-point o?set was taken into account.
In light of the ambiguous results above,it is worth returning to the question of what degree of stellar depletion is consistent with the T94data.Because the RNB T e?errors are signi?cantly smaller than the T94errors,the average error using the T94equivalent widths is reduced to0.06dex.This provides a smaller but more precise sample than the one used in PWSN.
Adopting the T94equivalent widths instead of the RNB equivalent widths produces a signi?cantly di?erent cumulative distribution.We compare the observed distribution with theoretical simulations convolved with aσ=0.06dex gaussian in Figure8.The s0.3case
with0.32dex depletion is now the best?t,while the0.18and0.5dex depletion cases are only marginally consistent.We include this to emphasize that the di?erences between the observational data sets needs to be reconciled in order to set more precise bounds on stellar depletion.
In conclusion,we?nd that the large di?erence between the results of RNB and T94 in their dispersion analyses can be traced directly to equivalent width measurements.The overall deviations exceed those predicted from the quoted errors.There are signi?cant zero-point o?sets,and external comparison with a small overlap sample from SS does not clearly identify a problem with the T94values.We therefore caution that further observational work is likely needed to uncover the origin of the di?erences,particularly since the overall conclusions depend sensitively on the presence or absence of a small number of outliers.
5.Discussion
Knowledge of the primordial lithium abundance sets interesting constraints on Big Bang Nucleosynthesis.However,the determination of the primordial lithium abundance relies both on observational data as well as on the model for stellar depletion.The most recent7Li abundance data sets exhibit a core with little internal scatter and a small number of outliers; these properties have been used to argue that there is little,if any,room for any stellar depletion.We have analyzed the RNB data set,and?nd that with or without accounting for a trend with metal abundance the data is consistent with mild stellar depletion;the best ?t depletion is in the range of0.1??0.2dex.Theoretical models with rotational mixing depletion this low predict a core with small scatter,since the large majority of young stars have low,and similar,rotation rates.Therefore,the number of outliers is a stronger test of the presence or absence of dispersion from rotational mixing.Either the no depletion case or models with lithium as depleted as0.5dex are unlikely based on both KS tests and the predicted number of overdepleted outliers as compared with the observed number.Our results di?er from those of RNB because they detrended the data in the log(Li)??Log(Fe) plane rather than in the linear Li??linear Fe plane which is more appropriate when testing for the presence of post-BBN7Li production.Similar conclusions can be derived from the observed ratio of6Li to7Li.
We have also compared the T94and RNB data sets,and?nd that the di?erent con-clusions that the two papers drew about the dispersion in lithium among halo stars can be traced directly to di?erences in equivalent width.If the T94equivalent widths are used instead of the RNB equivalent widths,a stellar depletion factor of0.3dex is inferred.We ?nd no compelling evidence of problems in the T94data by comparing both it and RNB
with an(admittedly small)set of stars in common with other studies.This indicates that there is further work to do on the observational front before making sweeping claims with implications for cosmology.Systematic observational errors from the temperature scale(0.05 dex),choice of model atmospheres(0.08dex),and equivalent width zero-point errors(0.05 dex)alone yield an uncertainty of0.11dex in the observed7Li abundance before considering stellar depletion??an error of similar magnitude to the theoretical uncertainties.In the last two sections we discuss the issues and uncertainties involved in the stellar modelling and the implications for BBN.
5.1.Stellar Physics Models
Further improvements are also desirable in the theoretical modelling of mixing and di?usion processes in the envelopes of low mass stars.These fall into the general categories of improved stellar physics on the one hand and better knowledge of the distribution of initial conditions and the angular momentum loss law on the other hand.
In Population I stars we have extensive empirical data on the distribution of rotation rates as a function of mass and age.We can only observe very metal poor stars when they are old,and therefore must extrapolate the behavior of Population I stars into a di?erent metallicity regime.The best prospect for constraining the theory is in observations of young clusters of intermediate metallicity;this will permit a direct test of the distribution of rotation rates and their time evolution.If the fraction of rapid rotators is di?erent from that present in Population I open clusters,the predicted number of outliers would be a?ected.More e?cient angular momentum loss in metal-poor stars could also reduce the predicted number of overdepleted stars for a given absolute depletion.
On the stellar physics side,the important uncertainties are internal angular momentum transport and the interaction of gravitational settling and rotational mixing.Helioseismic data indicates that the solar internal rotation is independent of depth in the radiative core down to0.2solar radii(e.g.Chaplin et al.(1999));the situation in deeper layers is less certain(compare Chaplin et al.(1999)with Gavryuseva,Gavryusev,&di Mauro(2000)). The spindown of young open cluster stars,however,is not consistent with uniform rotation enforced on a very short timescale(Krishnamurthi et al.(1997)).This combination implies that the timescale for e?ective angular momentum coupling between the surface and interior is intermediate between the ages of the young open clusters(50-100Myr)and the Sun(4.57 Gyr.)We are currently evaluating models in the limiting case of uniform rotation at all times to infer the impact on the predicted depletion.The general sense would be to reduce lithium depletion in models with shallower convection zones(because the di?usion coe?cients are
larger if the core rotates more rapidly than the surface.)Therefore the predicted degree of depletion in halo stars for a given solar calibration will be reduced.However,a range of stellar depletion factors for a range in solar initial conditions will still be possible;the net e?ect will be to make the observed halo star depletion consistent with less extreme values of the solar initial conditions.
Finally,gravitational settling and microscopic di?usion could a?ect surface lithium abundances.The gravitational settling of helium will produce a mean molecular weight gradient below the surface convection zone;composition gradients could reduce the e?ect of mixing(see Zahn(1992)for a discussion.)If rotational mixing was simply suppressed,how-ever,models with gravitational settling predict a decrease in halo star surface lithium with increased e?ective temperature which is not observed(Chaboyer et al.(1992)).Vauclair (1999)has raised the possibility of a nonlinear interaction that results in the suppression of both mixing and di?usion.This is an interesting possibility that should be investigated. There are,however,some factors that make a complete cancellation unlikely in our view.
First,the physical conditions in Population I stars with temperatures similar to the plateau stars are not very di?erent from the halo star conditions.We observe strong depletion and dispersion in M67stars with temperatures around6200K,which suggests that mixing is not inhibited in solar abundance stars where the timescale for gravitational settling is similar to the halo star case.In addition,Chaboyer,Demarque,&Pinsonneault(1995)investigated the interaction of gravitational settling and mixing for an earlier generation of models.They found that the time and mass dependence of lithium depletion was di?cult to reconcile with a strong suppression of mixing by settling.It is important to test the interaction of these physical processes against Population I data.In addition to the lithium question,such models must address the apparent absence of a gravitational settling signature in the turno?region of globular clusters(Chaboyer et al.(1992);Bergbush&VandenBerg(2001)).
5.2.Implications for BBN
Uncertainties in lithium equivalent width measurements,temperature scales,and model atmospheres introduce a systematic error in the determination of lithium abundances(on the log scale)which we estimate as±0.1dex,in agreement with Ryan et al.(2000)’s detailed analysis of the error budget.Re?ecting this uncertainty,the level of the Spite plateau, before accounting for depletion by rotational mixing or for post-BBN lithium production, has been variously estimated as2.1(RNB),2.2(Bonifacio&Molaro(1997),Bonifacio, Molaro,&Pasquini(1997)),and2.3(T94).We adopt2.2±0.1for our baseline estimate. We have found that the residual dispersion in the RNB data is well accounted for in a model