FYSAST

Examensarbete 15 hpNovember 2010

Atomic Diffusion in Old Stars

Testing parameter degeneracies

Thomas Nordlander

Institutionen för fysik och astronomiDepartment of Physics and Astronomy

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Atomic Diffusion in Old Stars

Thomas Nordlander

The predicted primordial lithium abundance differs from observations of unevolvedhalo stars on the Spite plateau by a factor two to three. Surface depletion due toatomic diffusion has been suggested as a cause of this so-called cosmological lithiumproblem. Evolutionary abundance trends indicative of atomic diffusion have previouslybeen identified in the metal-poor globular cluster NGC 6397 ([Fe/H] = -2), withstellar parameters deduced spectroscopically in a self-consistent manner. Abundancesof five elements (Li, Mg, Ca, Ti, and Fe) were found to be in agreement with stellarstructure models including the effects of atomic diffusion and a free-parameterdescription of turbulent mixing at the lowest efficiency compatible with the flatness ofthe Spite plateau.

It is our aim to evaluate the interplay of modelling assumptions and theoreticalpredictions under various priors, e.g. the independent age determination using thewhite dwarf cooling sequence, and the high efficiency of turbulent mixing recentlyfound compatible with halo field stars.

We perform self-consistent spectroscopic abundance analyses at an expandedeffective temperature scale inspired by results of new photometric calibrations fromthe infrared flux method. The resulting abundances are compared to predictions in agrid of theoretical isochrones, chosen in light of the priors for age and efficiency ofturbulent mixing.

We find that the observed abundance trends are not artefacts of the effectivetemperature scale, as it cannot be arbitrarily modified to flatten all trends. Theinferred abundance trends seem to be in agreement with predictions for an agecompatible with the white dwarf cooling sequence, and a limited range of weakturbulent mixing. The inferred initial lithium abundance of these stars is merely 30 %lower than the primordial abundance, discrepant at 1.5 standard deviations. Hence, astellar solution to the cosmological lithium problem is still within reach.

UPTEC FRIST08 000Examinator: Bengt EdvardssonÄmnesgranskare: Bengt EdvardssonHandledare: Andreas Korn

Sammanfattning

Vid Big Bang sammanfogades protoner (som utgör kärnan i väteatomer) ochneutroner till helium och mindre mängder litium. Halterna går att beräkna,men det finns inget sätt att direkt observera huruvida resultatet stämmer –man måste alltid tillämpa en övergripande modell av kosmos, och en specifikmodell som beskriver just det som observerats, exempelvis stjärnors atmosfärer.

Metaller, det vill säga ämnen tyngre än helium, byggs upp i tunga stjär-nor och sprids i galaxen av supernovor och stjärnvindar. Ju längre bakåt itiden man går, desto lägre halt av metaller bör man därför finna. Mycket me-tallfattiga stjärnor (en metallhalt som är en hundradel av solens) förväntasdärför vara mycket gamla, och hos dessa finner man i undersökningar sedan1980-talet praktiskt taget alltid samma halt litium (ungefär en litiumatom perfem miljarder väteatomer). Eftersom de observerade stjärnorna visar variationi yttemperatur och massa är det långt ifrån självklart vad för slags processsom skulle kunna ändra litiumhalten precis lika mycket för alla dessa stjärnor,så att litiumhalten antingen ökas eller sänks från värdet vid Big Bang till detobserverade. Det hela blev än mer problematiskt tjugo år senare, när satellitenWMAP lämnat data som ihop med modeller förutspådde en litiumhalt två–tregånger högre än den observerade. Oförmågan att förklara denna skillnad kallasdet kosmologiska litiumproblemet.

Modeller som beskriver stjärnors utveckling bygger på en rad fysikaliskaantaganden, och vissa förenklingar. En av dessa förenklingar är att man igno-rerar diffusionsprocesser, där olika ämnen genom sina mikroskopiska rörelsertillåts göra en netto-vandring genom stjärnan med hjälp av någon fysikaliskmekanism. Gravitationen får alla ämnen tyngre än väte att sjunka mot bot-ten (liksom kaffepulver tenderar att sjunka till botten). En gradient (övergång)i koncentrationerna tenderar att utjämnas av atomernas slumpvandring (endroppe mjölk i kaffet sprids i hela koppen). Om man samlar alla dessa meka-nismer finner man att effekten blir alldeles för stor, enligt jämförelser mellan dehalter som observeras i solen och dem man finner i meteoriter som bildades sam-tidigt som solsystemet. Lösningen är att lägga på en godtycklig faktor tänkt attmotsvara turbulens, som kraftigare motverkar gradienter i koncentrationerna.

Ett typiskt beteende för atomisk diffusion är den gravitationella effekten.Vissa ämnen tenderar att sjunka inåt, så att deras ämneshalt vid stjärnansytlager minskar. Ju äldre stjärnan är, desto längre tid har den haft på sig attlåta ämnen sjunka inåt. När kärnreaktioner i stjärnans centrum förbrukat dentillgängliga vätgasen börjar dess atmosfär expandera, så att den över en miljardår blir till en röd jätte. Under denna omvandling sätter konvektion – storskaligomblandning – in i atmosfären, så att de ämnen som tidigare sjunkit blandasupp i ytlagren igen.

Eftersom endast stjärnans ytlager är observerbara måste man därför under-söka likartade stjärnor både innan och efter att atmosfären utfört denna om-blandning. Skillnaden i ämneshalter utgör en trend, och påvisar hur stor mängdsom dolts i stjärnans djupare lager. Svårigheten ligger i hur man uppskattar

ämneshalterna. Den vanliga metoden kallas spektroskopi, där man delar uppljuset från en stjärna i dess färger – ett spektrum – och från detta härleder stjär-nans egenskaper och ämneshalter. Ämneshalterna kan inte observeras direkt,utan måste härledas ur modeller som återskapar det observerade spektrumet.

Den spektroskopiska metoden bygger på att varje atomövergång sker vid enviss energi, och varje sådan energi motsvarar en viss våglängd hos ljus – fotonerav viss energi. Atomövergången interagerar med fotoner av rätt våglängd, såatt ljus vid just den gällande våglängden saknas i stjärnans spektrum – dettakallas en spektrallinje. Mängden saknat ljus, spektrallinjens styrka, svarar motämneshalten samt den kvantmekaniska sannolikheten för att övergången skallske. Detaljerade resultat fås med hjälp av modeller som beskriver hur spektral-linjerna bildas, under inmatning av stjärnans yttemperatur och ytacceleration(g) – en felaktig temperatur ger felaktiga ämneshalter.

Man kan spektroskopiskt bestämma temperaturen genom att gå baklänges:vilken temperatur behövs för att en känd ämneshalt ska ge rätt spektrallinje?Denna teknik tillämpas på vätelinjer, där ämneshalten är känd. Men yttem-peraturer kan även uppskattas genom att jämföra med andra, väl undersöktastjärnor.

I den artikel varpå denna undersökning bygger användes den spektroskopis-ka metoden konsekvent. Man undersökte 18 stjärnor i en metallfattig stjärnhoppå ungefär 8000 ljusårs avstånd, och fann ämnestrender som med hög sanno-likhet antyder att diffusionsprocesser varit aktiva.

Vid senare jämförelser med väl undersökta stjärnor har det visat sig att despektroskopiskt uppskattade yttemperaturerna är något för låga. Denna studiebygger därför på samspelet mellan de uppskattade yttemperaturerna och vilkaämneshalter och trender de leder till. Vi har även undersökt hur dessa passarteoretiska förutsägelser då olika kraftig turbulens och olika ålder hos stjärnornaantagits.

Vi finner att man inte kan manipulera yttemperaturerna för att få äm-neshalternas trender att försvinna. Under den nya uppskattningen av yttem-peraturer hos stjärnorna fås något svagare trender, som verkar passa väl ihopmed teoretiska förutsägelser för en stjärnhop vid 11,5 miljarder års ålder. Dettaresultat överensstämmer med en tidigare uppskattning av stjärnhopens ålder,som använt observationer av mycket ljussvaga vita dvärgar. Vi kan inte slut-giltigt säga hur pass bra förutsägelserna överensstämmer, då vår uppsättningteoretiska modeller inte var tillräckligt väl vald – just där resultaten verkarpassa bäst har vi ingen modell att jämföra med.

Vår härledda ursprungliga litiumhalt avviker med ungefär 45 % från denförutspådda halten (istället för de klassiska 100− 200 %), en avvikelse av vissstatistisk betydelse. Denna skillnad förväntas minska något – kanske ända till25 % – när man tar hänsyn till mer avancerade modeller för linjebildning. Deslutgiltiga detaljerna beror på just vilken teoretisk modell som visar sig passaresultaten bäst, men det handlar inte om några omvälvande skillnader.

Contents

1 Introduction 91.1 The cosmological lithium problem . . . . . . . . . . . . . . . . 101.2 Globular clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Theory 132.1 Stellar models with atomic diffusion . . . . . . . . . . . . . . . 132.2 Stellar model atmospheres . . . . . . . . . . . . . . . . . . . . . 162.3 Spectral line formation . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 The strength of a spectral line . . . . . . . . . . . . . . 172.3.2 Spectral line broadening . . . . . . . . . . . . . . . . . . 182.3.3 The application of spectral lines . . . . . . . . . . . . . 18

2.4 Relaxing modelling assumptions – 3D and NLTE . . . . . . . . 192.4.1 Hydrodynamical models – 3D . . . . . . . . . . . . . . . 202.4.2 NLTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.3 NLTE mechanisms . . . . . . . . . . . . . . . . . . . . . 22

2.5 Spectroscopic observations . . . . . . . . . . . . . . . . . . . . . 232.6 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Background 293.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 3D and NLTE corrections . . . . . . . . . . . . . . . . . . . . . 303.3 The original temperature scale . . . . . . . . . . . . . . . . . . 32

3.3.1 Anticorrelations and peculiarities . . . . . . . . . . . . . 323.3.2 Abundance trends . . . . . . . . . . . . . . . . . . . . . 343.3.3 Comparison to diffusion models . . . . . . . . . . . . . . 353.3.4 Comparison to photometry . . . . . . . . . . . . . . . . 36

4 Results 394.1 Examining different temperature scales . . . . . . . . . . . . . . 394.2 Examining an expanded temperature scale . . . . . . . . . . . . 42

4.2.1 Abundance trends . . . . . . . . . . . . . . . . . . . . . 424.2.2 Comparison to diffusion models . . . . . . . . . . . . . . 43

4.3 Attempting to remove the diffusion signature . . . . . . . . . . 434.3.1 Abundance trends . . . . . . . . . . . . . . . . . . . . . 444.3.2 Comparison to diffusion models . . . . . . . . . . . . . . 44

4.4 Chromium and nickel . . . . . . . . . . . . . . . . . . . . . . . . 444.4.1 NLTE and 3D . . . . . . . . . . . . . . . . . . . . . . . 46

5 Conclusions 475.1 The original temperature scale . . . . . . . . . . . . . . . . . . 475.2 The extended temperature scale . . . . . . . . . . . . . . . . . . 485.3 Attempting to remove the diffusion signature . . . . . . . . . . 505.4 Implications for the primordial lithium abundance . . . . . . . 50

6 Summary 536.1 Ongoing and suggested future efforts . . . . . . . . . . . . . . . 53

A Abundance trends using the ionization equilibrium 55A.1 Examining an expanded temperature scale . . . . . . . . . . . . 55A.2 Attempting to remove the diffusion signature . . . . . . . . . . 55

B Tables and figures 57

References 65

Glossary 69

Nomenclature 73

Chapter 1

Introduction

This work is an extension of the previous investigation of Korn et al. (2007),pertaining to the effects of atomic diffusion in old stars.

Its basis lies with the cosmological lithium problem. In essence, Spite andSpite (1982) find a roughly constant lithium abundance among warm, metal-poor (old) halo stars. From the view of galactic chemical evolution wheremetallicity increases with time, this should straightforwardly correspond tothe primordial lithium abundance, and this was the favored interpretation forroughly twenty years. Calculations of Big Bang Nucleosynthesis coupled withWMAP cosmological data now suggest a primordial lithium abundance a fac-tor of two to three greater. Thus, the cosmological lithium problem: observa-tions strongly disagree with predictions, suggesting a serious error in standardcosmology, fundamental physics, or stellar physics. However, long before thecosmological lithium problem was realized, Michaud et al. (1984) found thatthe effects of atomic diffusion, traditionally neglected in stellar models, shouldcorrespond to a factor two or more depletion from the formation abundance ofthese stars. Thus, this mechanism could explain the discrepancy.

Korn et al. (2007) confirm this finding in an investigation of 18 stars atvarying evolutionary state (between the turnoff point and the red giant branch)in the metal-poor globular cluster NGC 6397. The observational signatures ofatomic diffusion are detected at statistical significance up to 3σ in five elements;magnesium, calcium, titanium, iron, and lithium itself – shown in comparisonto theoretical predictions in Figure 1.1. From the comparison of theoretical pre-dictions to observations, an initial lithium abundance of log ε(Li) = 2.54± 0.1is found to agree within the 1σ error bars with the then predicted primordialabundance log ε(Li) = 2.64 ± 0.03 – the logarithmic 0.1 dex difference corre-sponds to just 25 %, significantly less than the cosmological lithium problem.

However, new data from WMAP and nuclear physics raise the primordialabundance to log ε(Li) = 2.71 ± 0.06. Additionally, the stellar abundancescannot be directly observed, but require modelling. Using sophisticated non-LTE effects, the observed abundances decrease to log ε(Li) = 2.48± 0.1. Thus,the cosmological lithium problem returns as the discrepancy grows to 70 %, adisagreement at the 2σ confidence level.

Thus, it seemed appropriate to reexamine the previous results in the light ofnew findings. For instance, updated photometric calibrations suggest higher ef-fective temperatures for the turnoff-point stars than those derived spectroscop-ically. Higher effective temperatures weaken lines of minority species, e.g. theinvestigated lithium line, leading to a greater abundance. However, changingthe assumed effective temperature would affect all observed elements, requiringa proper reexamination of the effects.

Additionally, the precise effects of atomic diffusion depend on the strength

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Figure 1.1: Comparison of observations to predictions of diffusion models fora cluster age of 13.5 Gyr. Tx.x is shorthand for isochrones using turbulentmixing parameter logT0 = x.x. The T6.0 isochrone is found to match obser-vations well at the temperature scale introduced by Korn et al. (2007). Here,abundance trends are incompatible with traditional stellar models, which in-dicate a constant abundance for each element except lithium, where dilutioneffects set in at the SGB.

of turbulent mixing, for which only a free-parameter description exists. Theprocess is not yet properly understood, so within the framework of the currentdescription of turbulent mixing, one must find the range of the free parameterin concordance with observations.

This work extends the investigation of Korn et al. (2007) by examining theabundances on two new temperature scales: one suggested by photometry, andone designed to remove the abundance trends. We attempt to extend the ana-lysis with additional elements, but due to modelling shortcomings cannot drawconclusions from their behavior. Finally, we attempt to determine whether anexternal determination of the cluster age (Hansen et al., 2007) is compatiblewith the abundance trends, and reexamine the implications for the cosmologicallithium problem.

The cosmological lithium problem, galactic chemical evolution, and the ap-plications and shortcomings of globular clusters are reviewed in Section 1.1 andSection 1.2. The process of atomic diffusion, stellar modelling, and the methodof spectroscopy are detailed in Chapter 2. The investigation of Korn et al.(2007) is reexamined in Chapter 3, and our new investigation is detailed inChapter 4. The results are evaluated in Chapter 5.

1.1 The cosmological lithium problemWith a primordial mix of only the hydrogen, helium, and lithium created byBig Bang Nucleosynthesis a few minutes after the Big Bang, generations ofstars have enriched the gas content of galaxies later on. The gas content of agalaxy is known as the interstellar medium, and its content of elements heavier

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than helium (metals) is known as the metallicity.During the lifetime of a star, the main thermonuclear processes involve

the conversion of hydrogen into helium in the stellar core. As stars evolveinto the giant stages, they also form metals in the core, mainly ranging fromcarbon to iron, with details depending mainly on the stellar mass. Due to thecharacteristics of the thermonuclear reactions, the stellar mass also stronglycontrols the rate of evolution. Lifetimes range from roughly ten billion yearsfor stars similar to the Sun, to just millions for stars a hundred times moremassive. Stars less massive than the Sun are expected to have lifetimes longerthan the age of the Universe.

As the stellar core depletes the available hydrogen, the nuclear reaction ratesdecrease, and the decreasing core luminosity causes it to contract. By conser-vation of entropy, the stellar atmosphere expands, causing the star to leave thedwarf state on the main sequence and turn into a giant. As the atmosphereexpands, convection causes mixing in a zone reaching increasingly deeper. Aslayers containing processed core material are reached, the first dredge-up oc-curs when these elements mix into observable layers. Later, stellar winds expelparts of the outer atmosphere, with whatever elements may be present. Addi-tionally, a sufficiently massive star will experience the supernova process, whereit rapidly creates and destroys metals in great quantities, and expels at least alarge fraction of them. This is the process of chemical enrichment.

As the nuclear processes are limited to the stellar core, surface metallicitycould be expected to stay unaffected during the star’s lifetime, at least until itturns toward the giant stage. Its surface metallicity should match that of thematerial from which the star formed. Assuming rapid mixture processes result-ing in a homogeneous interstellar medium, one would imagine that stars withsurface compositions of lower metallicity formed from an interstellar mediumof low metallicity, indicating a high age for the star. As a rough indicator fora star’s age, this does not seem inconsistent with the idea of short lifetimesfor massive stars – observations of the most metal-poor stars do indicate lowmasses and enrichments by massive supernovae, seen as α-enhancements, i.e.enhanced abundances of elements built from α-particles.

Observations of unevolved metal-poor stars shows a curious result knownas the Spite plateau: the lithium abundance barely changes with stellar param-eters (Spite and Spite, 1982). The naıve interpretation tells us that this mustbe the primordial lithium abundance formed by Big Bang Nucleosynthesis,unaffected by galactic chemical evolution and stellar processes.

A problem with this interpretation emerged later, when calculations ofBig Bang Nucleosynthesis were coupled with the cosmological data gatheredby WMAP – specifically the factor ηB ≡ nB/nγ , i.e. the number ratio ofbaryons to photons. The results were a primordial lithium abundance on thelogarithmic scale of log ε(Li) ≈ 2.64 ± 0.03, which compares to the observedSpite plateau at log ε(Li) ≈ 2.3 by a factor of 2–3 (Spergel et al., 2007).Later data from WMAP along with updated nuclear cross-sections have ex-acerbated the problem by raising the predicted primordial lithium abundanceto log ε(Li) = 2.71 ± 0.06 – an increase by ∼ 20 % (Cyburt et al., 2010).Steigman (2010) includes additional observational data in the modelling. Thestandard Big Bang Nucleosynthesis model coupled with observed deuteriumabundances, and the non-standard model with additional equivalent neutrinoflavors, indicate log ε(Li) = 2.65± 0.06 and log ε(Li) = 2.66± 0.06 respectively.We adopt the WMAP-only value log ε(Li) = 2.71± 0.06 in our analysis.

Observationally, follow-up analyses of stellar samples find a lithium abun-dance slightly dependent on mass (Melendez et al., 2010) or metallicity (Ryan

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et al., 1999; Sbordone et al., 2010). Obviously, something must be wrong withour understanding of either fundamental physics, cosmology, or galactic orstellar evolution.

Atomic diffusion allows lithium to settle toward depths where it is heavilydepleted, as suggested by Michaud et al. (1984) and identified observationallyas a solution to the problem by Korn et al. (2006b, 2007) – upon which thiswork is based. It involves measurements of stars at different evolutionary stagesin a globular cluster (see Section 1.2), with comparisons to predictions of stellarmodels incorporating the relevant mechanisms. The theory involved is reviewedin Section 2.1, and we revisit the results concerning lithium in Section 5.4.

1.2 Globular clustersGlobular clusters are dense collections of stars formed within a galaxy, withnearly the same age and composition, as well as the same distance to us. Inthis picture, they are known as simple stellar populations – globular clusterstars at varying evolutionary stages should differ only in their initial mass,which determines the other attributes of the star. A more massive star willtypically be more luminous, larger and hotter at a given evolutionary stage,and due to the interplay of these factors it will evolve more rapidly.

As the globular cluster stars form, their metallicity mirrors locally thatof the galaxy at the time, which roughly indicates their age. However, thecluster stars are expected to form over an extended period, which allows thevery most massive and short-lived stars to enrich (pollute) their surroundingsbefore all cluster stars have formed. The intracluster gas (corresponding tothe interstellar gas of galaxies) from which the stars form can be thought tomix sufficiently rapidly that the composition of a specific star depends strictlyon when it formed, as compared to the complete stellar formation history ofthe cluster – similar to the naıve view of stellar formation in a galaxy. Thecluster thus contains both first and second-generation stars, formed from eithera pristine or polluted composition.

The process of pollution can be thought of as a small-scale version of galacticchemical evolution. Rather than completely shaping the metal content of thesecond generation, mere details are shifted, due to the limited time available.Additionally, following Gratton et al. (2004), one could expect stars of the firstgeneration to during their lifetime accrete polluted material from companionsor nearby stars. In the accretion scenario, one would expect the signature ofpollution to disappear from the surface as the star evolves into stages with deepconvection. In the formation scenario, this would not occur, as the star has ahomogeneously polluted composition.

For instance, AGB stars experience hot bottom burning where the corereaches convective layers, causing oxygen to be depleted in the CNO-cycle, andmagnesium and neon respectively to deplete as they form aluminum and sodiumvia proton-capture. These relations are seen as anticorrelations for O-Na andMg-Al, detailed in e.g. Gratton et al. (2001).

The O-Na anticorrelation has been identified by Carretta et al. (2009)amongst the majority of RGB stars in a sample of 15 globular clusters – forNGC 6397 the first and second generations are found in a roughly 1:3 ratio.

Typically, a more massive globular cluster is better at retaining its intra-cluster medium once pollution has commenced, while less massive clusters riskinstead having it ejected before pollution takes place. Additionally, less mas-sive clusters – like NGC 6397 – can be expected to contain fewer very massivestars, giving a lower overall expected effect of pollution.

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Chapter 2

Theory

2.1 Stellar models with atomic diffusionTraditional stellar structure models solve coupled differential equations. Asimplified description, as given by Prialnik (2010, equations 5.1–5.4), considers:

• The equation of hydrostatic equilibrium, where pressure balances gravity.

• The continuity equation, describing the distribution of mass.

• The thermal equilibrium equation, stating that energy produced withinis transfered outward.

• The radiative transfer equation, giving the temperature distribution fromthe flow of energy through partially opaque matter.

Stellar models with atomic diffusion add to the traditional equations atreatment of how microscopic movements result in a net transport of elementsthrough the star. Its effects are due to several mechanisms, described in asimplified manner by Richard et al. (2005, equation 5):

• Gravitational settling, causing heavier elements to be pulled toward thecenter, and descend into the star.

• Radiative levitation/acceleration, hindering the descent for ”large” ele-ments, or even lifting them.

• Thermal diffusion.

• Net movement due to concentration gradients.

• Turbulent mixing, counteracting concentration gradients by means of ad-ditional mixing.

The first four are determined from first principles, in a self-consistent way.Turbulent mixing is described by a parametrization of processes dominat-

ing below the convection zone, with details constrained by observations. Theturbulent mixing coefficient is given by Richard et al. (2005) as

DT = 400DHe(T0)

[ρ

ρ(T0)

]−3

(2.1)

where T0 is the free parameter, represented as a reference temperature, deter-mining the efficiency of turbulence. DHe(T0) is the helium atomic diffusioncoefficient at T0. The factor 400 arbitrarily determines the overall strength,

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while the exponent −3 determines the localization of the mixing, as calibratedby observations of the solar beryllium abundance being similar to the initialabundance. Lithium destruction occurs at layers where T ∼ 2.5 · 106 K (orlogT ∼ 6.4). By setting T0 < 2.5 · 106 K, turbulent mixing is restricted tothe outer layers, where temperatures are sufficiently low that depletion is min-imized.

In general, atomic diffusion causes heavy elements to diffuse toward greaterdepth. Details depend on specific properties of each element, when radiativelevitation competes with gravitational settling. At the surface layers, this isseen as a general depletion of metals. For low-mass, metal-poor stars, a thinconvection zone exists in the outermost regions, observable as granulation. Set-tled elements rest below this region. As the star reaches the giant stages, theconvection zone expands inward until the layers with enhanced metallicity arereached, causing the settled elements to reemerge. Later on, this includes ele-ments processed in the stellar core, causing the well-established first dredge-up.

An important effect on the star’s behavior is found when examining helium.As the second most abundant element, its settling in the deeper layers towardthe core causes an offset in the hydrogen abundance. This in turn causes nuclearreactions to more rapidly exhaust the core’s hydrogen, giving the star a morerapid evolution into the giant stages. An accelerated evolution for any givenstar translates into lower masses for the the stars of an isochrone, which in turnresults in a shift toward lower effective temperatures. Hence, the turnoff pointwill be found at a somewhat lower temperature (Richard et al., 2002).

Additionally, the settling of helium from the surface layers causes a changein the mean molecular weight, which corresponds to a shift in the surface gravity(Korn et al., 2007, Section 5.1).

The effect of turbulent mixing partially counteracts the effect of diffusion.Its presence below the convection zone hinders the downward diffusion of heavyelements. Lessening the effects of atomic diffusion, a more efficient turbulence(higher reference temperature T0) causes flattened diffusion trends. As turbu-lence acts only in the outer regions, helium settling in the core is not affected,and so the overall evolution will be similar regardless of this parameter. Thereare also exceptions to the general behavior of turbulence in the outer regions.With a sufficiently high efficiency of turbulence, settling is shifted toward thedeep, hot layers where lithium is destroyed.

With a non-destructive efficiency of turbulence, lithium is simply depositedwithin. As the convective zone expands inward, these layers are reached, andlithium resurfaces, giving an up-turn when comparing the SGB to the TOP.This causes an enhancement in lithium surface abundances at the subgiantstage. As convection runs deeper, it reaches layers depleted by thermonuclearreactions, causing the lithium surface abundance to drop according to the tradi-tional effect of dilution, just before the first dredge-up. With a higher efficiencyof turbulence, the deepening convection zone finds only depleted layers, insteadcausing the surface lithium abundance to drop smoothly, without the up-turn.

For other elements, the efficiency of turbulence mainly sets the depth of set-tling, which determines when the deepening convection zone causes resurfacingof deposited material, and how much is settled.

For calcium and titanium, radiative levitation may dominate over gravita-tional settling, unless suppressed by turbulent mixing. With a low efficiency ofturbulent mixing, these elements behave akin to lithium at a high efficiency ofturbulent mixing, with a slight down-turn at the subgiant stage as compared tothe turnoff point. This is due to radiative levitation hindering gravitational set-tling near the surface. Further into the star, the higher temperature results in

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ionization, which reduces the cross-section of the atom. Thus, these elementsare levitated near the surface, retaining the initial abundance, while deeperlayers experience gravitational settling, thus depleting. As the convection zonereaches these layers, dilution results in the lithium-like down-turn. With a highefficiency turbulence, the effect of radiative levitation is somewhat canceled.

The mass of a star determines its rate of evolution, and during the post-mainsequence evolution, the convective envelope grows. Additionally, the mass ofa star determines its structure. Notably, the temperature gradient determineswhether a layer is stable or affected by convection. The connection is that alower mass results in a thicker convective envelope. Combining these facts, wefind the thinnest convection zones for stars near the turnoff point – these arethe most massive, still unevolved stars.

Setting a lower age for an isochrone requires more rapid stellar evolution,corresponding to larger masses (and thus a shift towards higher temperatures).Hence, a lower age should correspond to thinner convective envelopes at theturnoff point. Additionally, a lower age leaves less time for atomic diffusion towork, which should straightforwardly result in flatter trends for all elements.One should however expect complications to this simple view. For instance,the different involvement of radiative levitation causes a possible non-trivialbehavior from the connection between age (mass) and luminosity. Add to thisthe varying thickness of the convective zone.

In summary, the expected effects of atomic diffusion are:

• A more rapidly progressing stellar evolution, with less massive and coolerstars at a given evolutionary stage.

• Generally lower abundances for non-temperature sensitive heavy elementsat early evolutionary stages, which return to nearly the initial values atthe late evolutionary stages.

• Stronger turbulent mixing causes flatter abundance trends when compar-ing evolutionary stages – the trends end up mostly similar to non-diffusivestellar models.

• A slight radiative levitation down-turn in abundances of relevant elements(calcium and titanium) at the SGB in a weak mixing scenario.

• An overall depletion of lithium, which is enhanced by stronger turbulentmixing. A slight up-turn in lithium abundance at the SGB in a weakmixing scenario, or extra depletion giving a down-turn in a high efficiencyturbulence scenario.

• Possibly flatter trends for a set of younger stars.

In light of this, a comparison to observations should be performed for a setof similar stars at varying evolutionary stages, for instance ranging from un-evolved stars to subgiants and red giants with the effects of dredge-up visible.Abundances of both lithium and other elements should be analyzed, as dif-ferent behaviors are expected from the same set of parameters. A sufficientlylarge such comparison should constrain both upper and lower limits to the freeparameter of turbulent mixing, or dismiss it altogether if these limits do notoverlap.

15

2.2 Stellar model atmospheresSpectroscopic observations of real stars give a connection to stellar models bymeans of line synthesis. This involves calculating a synthetic spectrum undersome assumed set of stellar parameters, relying on a model stellar atmosphere,and line formation theory. The latter is described in the next section.

The model stellar atmosphere is traditionally computed for a 1D relationbetween temperature T and a depth parameter – e.g. optical depth τ . For 3Ddescriptions, a single geometrical depth x simultaneously holds warm upflowsand cool downdrafts, and thus does not correspond to a single relation for Tand the depth parameter.

The computation of a model stellar atmosphere follows a similar recipe astraditional models of stellar structure, described in the previous section. Inour description, following Rutten (2003), we let subscripts i and j refer to anytwo atomic levels, while subscripts u and l refer to the specific upper and lowerlevels of a transition.

Under the LTE approximation, all processes are said to be local and inthermodynamical equilibrium locally, so ionization and excitation populationsare given directly by solving the Saha and Boltzmann equations. The non-LTEcase is described in Section 2.4.2.

The Boltzmann equation describes the ratio of any two excitational levelpopulations,

ninj

=gigj

exp[−χi − χj

kT

](2.2)

where excitational level i has population ni, statistical weight gi, and ex-citational energy χi relative to the ground level. The transition energy isχu − χl ≡ hc/λ, with λ the transition wavelength.

The Saha equation describes the ratio of populations in successive ionizationstages:

Nr+1

Nr=

1

Ne

2Ur+1

Ur

(2πmekT

h2

)3/2

exp[− I

kT

](2.3)

where ionization stage r has population Nr, partition function Ur, and ioniza-tion potential I. Ne is the number density of electrons, and me the electronmass. Combining these equations for all ionization stages of an element triviallygives the ratio of the population for an ionization stage to the total populationof the element.

Combined, the Saha and Boltzmann equations describe the population ofany excitational level at any ionization stage for an element. Supposing thatall partition functions and statistical weights are known (which they generallyare – many tabulations are available), a given temperature instantly solves forthe precise population distributions. This is the strength and point of LTEanalyses. See Section 2.4.2 for an overview of alternatives.

The radiative transfer equation is given as (following Gray, 2005)

dIνdτν

= Sν − Iν (2.4)

where Iν is the intensity, and τν the optical depth. Averaging the intensity overall space angles results in the mean intensity,

Jν ≡ 1

4π

∫IνdΩ. (2.5)

The source function depends on emission and absorption coefficients. Thesehave contributions from both the continuum and from individual spectral lines.

16

Under the diffusion approximation and LTE, Jν = Sν = Bν . Solving theradiative transfer equation (2.4) including all its constituents, and the equationof hydrostatic equilibrium, results in the distribution of temperature and partialgas pressures as a function of optical depth: the T − τ -relation.

2.3 Spectral line formation

For the behavior of spectral lines, we need to focus on the behavior of levelpopulations. Again following Rutten (2003), superscript l and c refer to vari-ables describing either the line or the continuum, and subscripts retain theirprevious meaning.

2.3.1 The strength of a spectral line

The Einstein coefficients describe the probabilities of three types of transitions.

• Spontaneous emission considers an undisturbed atom spontaneously de-exciting by means of photon emission. The relevant probability constantis Aul, as the transition connects an upper to a lower level.

• In the presence of radiation, stimulated emission (negative absorption)and true absorption (rather than scattering; also known as radiative ex-citation) are also possible, with probability constants Bul and Blu respec-tively. They are related as Bul = Blugl/gu, via the statistical weight g ofeach level.

• Collisional interactions have probability constants Cul and Clu.

The line source function affects the total source function in (2.4), and isgiven as

Slν =

jναν

=2hc

λ21

gunlglnu

− 1(2.6)

=2hc

λ21

exp[

hcλkT

]− 1

≡ Bν(T ) (2.7)

with, the statistical weights g and number densities n. jν and αν are the lineemission and extinction coefficients. While (2.6) is the general result, (2.7) istrue only in LTE by the assumption (2.2). Its result is known as the Planckfunction, which describes the spectral distribution of black-body radiation.

Line opacity is given by the line extinction coefficient. In LTE,

αlν =

hc

4πλ(nlBlu − nuBul). (2.8)

The strength of a spectral line is proportional to the comparison of the lineand continuum opacity,

R =αlν

κν(2.9)

where κν is given mainly by the continuum opacity of H− in cool stars.

17

2.3.2 Spectral line broadeningThe shape of a spectral line at a specific depth is described by a Voigt func-tion, which convolves Gaussian and Lorentzian (dispersion) profiles. For a suffi-ciently strong line, the Gaussian profile dominates the core while the Lorentzianprofile dominates the wings. Each is due to different processes, which broaden(damp) lines. The damping factor of a Lorentz profile is

ψ(ν − ν0) =γ

4π21

(ν − ν0)2 +( γ4π

)2 (2.10)

where ν is the investigated frequency, ν0 the (resonant) frequency of the tran-sition, and γ the damping constant.

Convolving Lorentzian profiles results in a broadened Lorentzian profile.The same holds for Gaussian profiles.

Natural broadening is an intrinsic effect, describing the uncertainty in en-ergy given by the Heisenberg relation. Its shape is a Lorentzian profile, with adamping constant given by the spontaneous decay probabilities (i.e. the inverselifetime) of the level.

Collisional broadening can be of four different types.The quadratic Stark effect and van der Waals broadening are relevant for

all elements. The former involves interactions between charged particles, andis important in hot stars. The latter dominates in cool stars, where collisionswith neutral atoms alter energy levels. Both have Lorentzian profiles.

Resonance broadening and the linear Stark effect are relevant for hydrogenatoms, by interactions with other hydrogen atoms or charged particles. Whileresonance broadening too has a Lorentzian profile, linear Stark broadening ismore complex (but becomes important only for hot stars).

Doppler broadening results from the velocity distribution of particles, giv-ing varying Doppler shifted energies. The usual Maxwell-Boltzmann velocitydistribution gives a Gaussian profile. Turbulent broadening is for simplicityseparated into microscopic and large-scale components, microturbulence andmacroturbulence, respectively. The effect of rotation is the same as for macro-turbulence.

2.3.3 The application of spectral linesThe final shape of the spectral line is determined by its full distribution offormation depths, where the varying local parameters determine the interplayof the source function and line opacity, as well as the different broadeningcomponents. Each observed wavelength of the spectral line has a different setof formation depths, as the line itself determines the optical depth. This set isknown as the contribution function.

In the core of a line, enough light is blocked that one cannot see veryfar into the atmosphere – the contribution function is thus dominated by thecooler outer regions of the atmosphere. In the wings, less light is blocked, sothe contribution function centers on deeper layers. Importantly, different partsof the line have different contribution functions. For a sufficiently strong line,the contribution function is shifted into the outer atmosphere where modelstellar atmospheres become less realistic (typically, they lack a chromosphere).Such lines cannot be completely described by the stellar model, limiting theirusefulness instead to the wings.

The equivalent width is a measure of the total line strength. The equiva-lent width of a spectral line is given by the width of a hypothetical, perfectlyrectangular, spectral line blocking the same amount of integrated light.

18

Altering the temperature shifts the ionization fraction in the Saha equation(2.3), causing a large relative population variation in the minority but not themajority species. Thus, minority species are temperature sensitive, as is thecontinuous opacity.

Altering the gravity changes the electron pressure Pe, i.e. Ne in our formof the Saha equation (2.3), again shifting the ionization fraction. For minorityspecies, the effect on line opacity cancels that of the continuous opacity (mainlycontributed by H−). Majority species however find only an altered continuousopacity. Thus, only majority species are pressure sensitive.

These mechanisms give a means of evaluating the effective temperature andsurface gravity of a star. If lines of different excitational energy show differentabundances, the excitational populations in the atmosphere do not match thoseof the model – thus, the wrong temperature (distribution) is being used. Forlines from different ionization stages of an element, incompatible abundancesindicate an incorrect surface gravity. These conditions are known respectivelyas the excitation and ionization equilibrium.

The effect of altered abundances is given by the curve of growth. Threetypical regions exist,

• Weak lines, where the Doppler core dominates. The equivalent width andline depth vary linearly with the number of absorbers (ε).

• Saturation, where the central depth asymptotically reaches the maximumvalue.

• Strong lines, where the broadening causes significant optical depth in theline wings. For cool stars, the pressure dependence of van der Waalsbroadening determines the behavior of the wings.

For a saturated line, broadening by microturbulence shifts part of the lineinto the wings, thus strengthening the line by avoiding the saturated wavelengthregion. For a weak line, only broadening is gained, as the line core is notsaturated.

The weak-line behavior of microturbulence is akin to the effect of macro-turbulence and rotation, which act only as external broadening profiles. TheDoppler shift of large-scale regions moving in different directions merely re-sult in a ”smeared” line, which is broadened but not strengthened. Thus, theequivalent width is hardly affected.

It seems a good idea to include the additional information contained in lineprofiles rather than blindly computing and comparing equivalent widths. Thecomparison of observed and synthesized line profiles can be performed manually,with comparison by eye. Lind et al. (2008, Figure 2) instead performs anautomated analysis, where the line shape is considered in a manner somewhatakin to the photometric method of measuring flux within a specified wavelengthregion. One could also imagine a setup where observed line shapes are evaluatedand corrected for spurious noise features and blending lines manually, beforean automated fitting procedure begins.

2.4 Relaxing modelling assumptions – 3D and NLTEWe give here a summary of the mechanisms and ideas involved in more advancedmodelling. A proper review is given by Asplund (2005), and mechanisms aredetailed in Rutten (2003). Understanding of these mechanisms is important asthey could possibly undermine any study not taking care to correct for theireffects.

19

2.4.1 Hydrodynamical models – 3D

3D models use physical descriptions of hydrodynamics to describe the processesin a stellar atmosphere, which results in granulation. Simply put, granulationdescribes an observable process of convection: warm upflows and cool down-drafts each have different temperature structures, giving components of severaldifferent line shapes, averaged into a shape which does not naturally arise from1D atmospheres. Instead, the free parameters micro- and macroturbulence aread-hoc varied to reproduce the spectrum. Inherent asymmetries of these mo-tions result in a slight line asymmetry which is not reproduced by free param-eters in 1D models. With warm upflows and cool downdrafts, the temperatureT and depth parameter, e.g. optical depth τ , at a single geometrical depth x isinhomogeneous. The radiative transfer equation (2.4) must be solved includingthese horizontal inhomogeneities. 3D models are thus, by their nature, quitecomputationally heavy.

An important result of 3D models is found for low-metallicity stars, wherethe low metallicity causes lower opacity. This decreases the effect of line heating(line blanketing), which leads to lower temperatures in the outer atmosphericregions. Heat transfer is then mainly adiabatic, rather than radiative. Theresulting temperature structure is thus steeper than in corresponding 1D at-mospheres. The cool outer layers in the steep temperature structure result inlarger populations for neutral minority species, making such lines stronger thanin 1D stellar atmospheres.

1.5D Due to the fact that 3D stellar atmospheres are computationally heavy,their treatment of line opacity is usually simplified. The level of sophisticationin NLTE analyses (described below) does not allow such simplifications. Onesolution is to improve the line opacity treatment in the 3D model, resulting inhighly realistic but challenging computations.

Alternatively, a 1D stellar atmosphere where the temperature distributionis gathered from the averaged results of a 3D simulation can be used in existingNLTE codes. Such an atmosphere is called 1.5D, as it somewhat retains thesuperior realism of the 3D atmosphere temperature distribution.

2.4.2 NLTE

LTE relies on interactions being local effects, i.e. a short mean-free path forphotons and collisionally dominated transitions. When this does not hold,reality can no longer be approximated by LTE – we must consider non-localeffects: non-LTE (NLTE).

These arise when radiative interactions dominate over collisions. For coolstars, metals rather than hydrogen act as the main electron donors. At suffi-ciently low metallicities, the decreasing electron pressure allows radiative pro-cesses to dominate (Gehren et al., 2004).

The following description of NLTE computations is based mainly on Rutten(2003). Level populations are not given by the Saha and Boltzmann equations(i.e. (2.3) and (2.2)), but rather by the statistical equilibrium equation,

0 =dnidt =

N∑j 6=i

njPji − ni

N∑j 6=i

Pij (2.11)

where zero equality represents the assumption of steady state, ni is the pop-ulation of level i, the transition rates are Pij = Aij + Bij Jν + Cij , with the

20

Einstein coefficients Aij for radiative emission, Bij radiative absorption (stim-ulated emission) and Cij collisional excitation (deexcitation), and Jν is themean intensity over all angles (2.5) as averaged for the transition.

The true populations are described by their resulting departure coefficients:

bi =nNLTEi

nLTEi

. (2.12)

These can be put into the NLTE form of the line source function, modifying(2.7). For bound-bound transitions,

Slν =

2hc

λ21

blbu

exp[

hcλkT

]− 1

≈ bublBν (2.13)

meaning that the result is a modified Planck function: if the lower level popula-tion suffers larger depletion than the upper level, the resulting source functionwill be stronger than the Planck function. This is known as superthermalradiation.

If however both level populations show similar departures, bu ≈ bl, thenSlν ≈ Bν . This is known as a near-LTE situation, and is found in e.g. Korn

et al. (2003) for Fe i.The line opacity (2.8) is also modified,

αlν =

hc

4πλbln

LTEl Blu

(1− bu

blexp

[− hc

kTλ

]), (2.14)

meaning that primarily the lower level population determines the line opacity.This means that even a near-LTE situation will give non-LTE line strengths.

In the radiative transfer equation (2.4), the intensity depends on the sourcefunction (2.13), which depends on the populations from (2.11) rather than theSaha and Boltzmann equations’ resulting Planck function (2.7). Thus, theintensity depends on the populations, which depend on the intensity. Suchanalyses cannot be restricted to a single transition, all transitions of a sin-gle ionization stage of an element, or even all transitions of all all ionizationstages of an element, as transitions of other elements may still also influencethese equations at the relevant wavelengths. Analyses where this is assumednonetheless are known as restricted NLTE, described further below.

Additional complexity is found with time dependent considerations, whichremove the zero-variation of the statistical populations (2.11), of importancefor hydrodynamical 3D stellar atmospheres.

An important note is that the levels and transitions required for the NLTEanalysis are not only determined by the lines used in the spectroscopic abun-dance determination. A sufficiently large difference in a few lines might resultin a species-wide effect.

Importantly, the lower ionization stage needs a more or less complete de-scription of the energy levels up to near the ionization potential - althoughclosely spaced excitation levels may be combined into superlevels to lessen thecomputational cost. This is important since a collisional coupling is requiredbetween the continuum electron reservoir and the highest excitation levels todescribe collisional ionization and recombination properly. The gap betweenthe highest excitational levels and the ionization potential should be on theorder of the typical kinetic energy, as given by a Maxwell-Boltzmann velocitydistribution (Mashonkina et al., 2010a).

If the highest energy levels are not sufficiently well described, the impact ofcollisional ionization will be lessened, causing exaggerated NLTE departures.

Corrections for NLTE effects can be applied in two ways:

21

• They can be given as a set of population departure coefficients bi. Thisis applied to the line formation routines as a correction from the LTEpopulations, giving the correct line shape during the spectroscopic partof the analysis.

• The resulting difference can be tabulated in a grid of corresponding valuesfor line strength and elemental abundance. From such a grid, one mayinterpolate to find the correction after performing a regular LTE analysis.

If line fits are used, the former method should be preferred as line shapes containmore information than what the simple after-the-fact correction provides. Thelatter method is however more powerful in that it may include additional effectsof 3D hydrodynamics, or a revised model atmosphere considering either of theseeffects.

The abundance correction for NLTE is always given as

∆ log ε = log εNLTE − log εLTE

with expressions of the same sign for corrections regarding 3D or 3D-NLTEmodelling.

Restricted NLTE The usual treatment of NLTE does not allow the deviantlevel populations to feedback on the model atmosphere – the stellar atmosphereis still calculated under the assumption of LTE, while line formation followsNLTE. This works under the assumption that one is dealing with a trace ele-ment, whose deviation from the LTE assumption will not heavily impact on thesurroundings. For instance, an ample electron donor would affect the overallelectron reservoir and thereby the continuous opacity. If this were the case,one would expect discrepancies between LTE and NLTE model atmospheres.

2.4.3 NLTE mechanismsOverionization In the deeper, hotter layers of the stellar atmosphere, thehigh temperatures cause the Planck function to shift toward shorter wave-lengths, with a rapid increase in the UV for cool stars. This steepness of thePlanck function creates a non-locality in the radiation field: the much greaterUV flux from deeper layers is able to seep into the colder layers. This isknown as superthermal radiation, since Jν > Bν . If a species has transitionswith bound-free transitions at UV energies, the superthermal UV radiation willcause greater photoionization than expected from LTE. Photoionization is thusthe main mechanism for overionization. Overionization amounts to a decreasedline opacity, making the lines weaker, which corresponds to an underestimatedabundance and thus a positive abundance correction. The species affected (forcool stars: the neutral stage of the element) will then suffer a depletion in thoseenergy levels, which if numerous will drain all populations of the species.

This effect is important for minority species, where a relatively small vari-ation in the populations of the majority species translates into a large relativechange. Therefore, since line opacities are sensitive to populations, they shouldbe sensitive to a variation in the amount of ionization. It is also enhancedby steeper temperature structures, since they connect to a steeper variation inthe Planck function. Use of a model atmosphere with insufficient steepness inthe temperature will thus result in NLTE masking (Rutten and Kostik, 1982).Importantly, the steeper temperature structures of 3D atmospheres will resultin stronger overionization effects (Asplund, 2005).

22

Photoionization of course depends on the cross-section for photoionizationof the lower energy level. Where such laboratory values are unavailable, ahydrogenic approximation is used.

Photon pumping Similar to overionization, but instead for excitationaltransitions. Transitions with energies in the UV region affected by superther-mal radiation will suffer excessive excitation, causing an overpopulated upperlevel. As the lower population depletes into the upper level, the source func-tion increases, causing a weakened line. The increase in the affected upperlevel also affects its other transitions, according to (2.13). Again, this validatesthe claim that the line list employed in a NLTE analysis needs to be more orless complete. The weakened line corresponds to an underestimation of theabundance.

Photon suction If a highly excited energy level, near the ionization poten-tial, is connected via a chain of radiative transitions all the way down to nearthe ground state, the so-called photon suction mechanism may become impor-tant. An atom in the ground state of the ionized stage could momentarilyrecombine with a free electron. With a sufficiently large transition probabil-ity compared to the photoionization probability, this mechanism could stepin before the atom is reionized, instead causing the electron to lose its excessenergy as a series of emitted photons. This results in under-ionization, wherethe neutral species has overpopulated levels. This causes excess line opacity,i.e. an overestimation of the abundance.

Resonance scattering With a sufficiently strong line, the opacity of the lineitself causes local depletion of radiation, Sν < Bν . This correction correspondsdirectly to a strengthened line, i.e. again an overestimation of the abundance.

2.5 Spectroscopic observationsConverting light as gathered by a telescope into a spectrum requires a disperser,known as a spectrograph. It uses a diffraction grating, composed of angledrulings (grooves) covered in a thin reflective layer, causing light to disperse bymeans of wavefunction interference.

The grating equation describes the distribution of interference maxima inthe dispersed light, (Gray, 2005, Chapter 3)

nλ

d= θ = sinα+ sinβ (2.15)

where α is the fixed angle of incidence between infalling light and the grating,β is the angle between grating and image, λ the wavelength of infalling light, dthe spacing of rulings, and n the integer order of the (symmetrical) interferencepattern counting from n = 0 at the central maximum. For a pointlike sourcecomposed of multiple wavelengths, the small relative shift in wavelength for aspecific interference order n will give a small shift of β, resulting in adjacentimages – a spectrum. For larger values of n – higher orders – the dispersionbecomes greater, corresponding to higher resolution. Gratings working at largen – on the order of 100 – are known as echelles.

Interference causes the formation of multiple maxima, but it is diffractionwhich determines their amplitude, following a sinusoidal which diminishes withincreasing interference order n – this is known as the diffraction envelope. This

23

is a problem, as large values of n give high resolution, but low values give astronger signal. Introducing a phase shift will affect only the diffractive be-havior, while retaining the interference pattern. Thus, the diffraction envelopeis shifted so that the largest amplitudes go to the desired large value of n forsome intended wavelength. The phase shift is set by the angle of the rulings inthe grating, and the method is called blazing.

With a blazed spectrum, the output is a single long line, which is projectedonto a photosensitive chip - a CCD. The higher the resolution, the longer thisline will become, thus making the observable bandwidth limited by the chipsize. However, this uses only a single line of a chip, which would need to beunreasonably wide. The solution involves cross-dispersing the spectrum usingadditional gratings, so that each order n projects along a line of the chip.Each order of the recorded spectrum has a blaze distribution which must becompensated. Failing to do so results in a ”wavy” continuum, which is difficultto compensate for in strong lines, of particular interest for effective temperaturedetermination using broad Balmer lines like Hα.

Having attempted to removed the blaze from each order, they are mergedinto a single continuous spectrum. As the orders overlap somewhat, there areno ”holes” in the spectrum. There may however remain blaze residuals fromthe individual orders. The wavelength range of an order n which does notoverlap the adjacent orders n ± 1 is known as the free spectral range. If awide line is not positioned within the free spectral range of any single order, itsbehavior can be affected more by artefacts of order merging than by the stellarparameters one wishes to investigate.

Examining faint stars, each exposure lasts roughly one hour. Often, severalsuch exposures must be combined to gain sufficiently high signal-to-noise ratio(S/N), which decreases by the square of the resolving power R. First, eachspectrum must be individually shifted to match a set of well-behaved and easilyidentifiable lines. This compensates for the varying radial velocity giving aDoppler shift – introduced by e.g. stellar motion, and Earth’s rotation andmotion – which would otherwise introduce duplicate or smeared lines.

The resolving power R ≡ λ/∆λ is chosen to conform to both observationalconstraints and physical requirements. R must be set so that ∆λ becomessufficiently small that adjacent lines be resolved (individual lines are typicallynot fully resolved), while at the same time giving a reasonably large S/N for theresulting spectrum to be useful. The former constraint is important in crowdedline regions, while the latter is important for the weakest lines or faintest stars.

2.6 Method

There are several methods for determining stellar parameters spectroscopically,using our knowledge from Section 2.3.

A common way of determining the effective temperature is by fitting theprofile of the hydrogen Balmer lines, primarily Hα. Due to stellar atmospheresbeing dominated by hydrogen, this line is very strong. This means that the coreforms in the outer layers above the photosphere, where our models no longerapply. Instead, one focuses on the wings, which do form in the photosphere.The atomic physics describing hydrogen is however quite difficult, and muchresearch has been devoted to it. For instance, Barklem et al. (2000) introduceshydrogen self-broadening terms from quantum mechanical calculations. Theseare however not applied in this analysis of Hα, which instead follows the res-onance broadening theory of Ali and Griem (1966), for consistency with Korn

24

et al. (2003). Korn et al. (2007) find typical differences of 20–40 K between thebroadening theories.

In lieu of complete 3D hydrodynamical atmospheres, the 1D hydrostaticatmospheres commonly used apply so-called fudge-factors to simulate hydro-dynamical effects: the microturbulence and macroturbulence. The microtur-bulence affects the strength of saturated lines, while weak lines are merelybroadened. Extending a weak line analysis to include strong lines, the cor-rect microturbulence is found when lines show an abundance independent ofline strength. Macroturbulence and rotation are applied as an external profile,often a Gaussian convolution which smears the lines without affecting theirstrength.

When fitting single lines in an abundance analysis, only the elemental abun-dance and external profile are varied, until the synthetic line matches the spec-trum in both depth and overall shape. The abundance found to match a singlespectral line is not an absolute measure. Due to uncertainties in the atomicdata (primarily the log gf value), different lines will generally indicate differentabundances. This problem is addressed by performing a line-by-line differentialanalysis: The same line is investigated in the solar spectrum, using a modelwith the standard solar parameters and identical atomic parameters. Dividingthe stellar abundance by the solar abundance derived for that line then givesthe differential abundance,

[X/H]i = log[ε(X)?,iε(X),i

]. (2.16)

If different atomic parameters are used, the line-by-line differential analyses issimply set to involve the log gf value, by comparing the solar measurementlog gfε,i to the stellar measurement log gfε?,i. Taking the mean of the abun-dances for each line, we find the stellar abundance of the element, with somerandom scatter. Supposing that the physics of the star is sufficiently similar tothe Sun, the systematic errors introduced by incorrect atomic parameters willby design cancel in a differential analysis.

An alternative to this type of line-by-line differential analysis, is an invertedanalysis of solar lines, described by e.g. King et al. (1998). Setting the bestknown solar element abundance, the atomic parameters are adjusted until themodel successfully recreates the solar spectrum. This cancels out the samesystematic effects as above.

Either way, there will still be other systematic errors. For instance, thecomparison of a metal-poor star to the Sun is deficient in the simple fact thatthe Sun is not a metal-poor star. Lines sufficiently strong to be detected ina metal-poor star will be so strong in the Sun as to be affected by differentformation mechanisms. We attempt to avoid this by including the weakestlines possible.

The method of finding the stellar parameters uses an iterative process ofthe above concepts.

1. Making an initial guess of the surface gravity, Balmer line line fittingusing e.g. Hα results in a corresponding effective temperature.

2. Using weak lines of neutral iron, i.e. the minority species, the iron abun-dance can be varied until agreement is found with individual line abun-dances in a line-by-line differential analysis.

3. Adding strong lines of neutral iron, the microturbulence is found whenindividual line abundances seem independent of line strength.

25

4. Adding lines of singly ionized iron, i.e. the majority species, the surfacegravity is varied until the same abundance is found for both species. Atthis point, the stellar parameters have been determined. If the result atthis point is found to be inconsistent with the Balmer lines, step 1 andforth should be reiterated until convergence is reached.

5. With the stellar parameters determined, elemental abundances are foundby varying only the relevant abundances separately for each line of theinvestigated element.

For certain elements, the available lines may be too weak for single star mea-surements – this is the case for sodium in our sample of TOP and SGB stars.This allows only rough estimates or upper limits to be set for single stars. Usingthe mean stellar parameters of each group, the group-averaged spectra of eachline might gain the S/N required for a precise abundance determination.

The resulting elemental abundances then carry a number of different pos-sible errors.

• Model discrepancies, e.g. employing an insufficient broadening theory foreffective temperature estimation from Balmer lines.

• Random scatter, noise, giving a false estimate of the line depth or shape.Additionally, noise might lead one to misjudge the continuum level, re-sulting in an offset normalization. This leads to lines being identified asbeing either too strong or too weak. For the elemental abundances, thistype of error is alleviated by taking the average of several spectra, causinga√N reduction of the noise.

• Unidentified blends are easily mistaken for noise. Careful analysis ofgroup averages where single noisy spectra have been discarded should helpdistinguishing between the two. If a blend is suspected but no suitableline found in the line list, one should compare the shapes of the spectralline for different evolutionary stages. If, for instance, the suspected blendconsists of an unexpected asymmetry in a spectral line of the dwarf stars,a comparison should be made to the giants’ spectra. In the giants, thehigher S/N along with the lower temperature (causing stronger lines inminority species) and surface gravity (stronger lines in majority species),should aid in their identification.

• Erroneous atomic data. Oscillator strengths give only a multiplicative fac-tor in the line strength, and so cancel out in a differential analysis. NLTEcalculations depend heavily on other atomic data, which are required fora larger number of levels and transitions to attain correct deviations.

• Erroneous stellar parameters. Two stars with a species showing identicallines would be found to have different abundances if the stellar parametersdiffered.

Having identified the errors, their magnitude must be estimated.

• The individual stellar uncertainty from scatter of individual lines is usefulfor elements with adequately numerous lines in the analysis, that thestatistical tools are not being abused. In our case, this is true only forour iron analysis, as all other elements use three or fewer lines.The scatter amongst individual stars then propagates the previous errorinto the error of the group-averaged result. The group-averaged iron

26

abundances range in uncertainty between 0.03 and 0.05 dex, dependenton the number of stars in the group.

• For the metals, group averaged abundances are estimated using groupaveraged spectra rather than an average of the individual abundances.As in the previous case, this assumes that the stars within a group arevery similar. The ”limits” to the abundance are then given by the rangein synthesized lines able to match the group averaged spectrum. With thetypical tip-to-tip S/N amplitude on the order 3 ∼ 4 standard deviationsfor a normal distribution, the typical 2σ estimation by this method forall but the weakest lines is 0.03 dex – an obvious underestimation. Itdoes however, in general, show the advantage of analyzing group-averagedspectra.

• From the uncertainty propagated from uncertainties in the stellar parame-ters. e.g. a typical uncertainty in the effective temperature ∆Teff ∼ 100 Kpropagates as roughly ∆ log ε ∼ 0.07 for Fe i.

In lieu of proper statistics for our small samples of metal lines, we assumetypical differential uncertainties of 0.05 dex for each element in each group. InSection 3.3 we mention the cases where star-to-star scatter is found to be largerthan this, and is thought to be real.

Using our above estimated errors, the significance of an observed diffu-sion signature is given by the discrepancy from the flat line predicted by non-diffusive stellar models, save for the case of lithium where the deepening con-vective envelope on the subgiant-giant branch causes a different behavior. Thetypical estimated errors of 0.05 dex, at both the TOP and RGB, propagate intoa standard deviation σ =

√σ21 + σ22 = 0.07 for the magnitude of the trend. We

define the measure of the trend for element X, unless stated otherwise, as

∆ log ε(X) ≡ log ε(X)RGB − log ε(X)TOP. (2.17)

27

Chapter 3

Background

We give here a short overview of the original investigation, published as Kornet al. (2006a,b,c, 2007).

Spectra were obtained for 18 stars in four groups of evolutionary states,indicated by crosses in Figure 3.1: five stars just past the turnoff point (TOP),two at the subgiant branch (SGB), five at the base of the red giant branch(bRGB), and six at the red giant branch (RGB). This was done as stellarmodels including atomic diffusion suggest a general settling of elements at theturnoff point, which gradually resurface as the stars evolve toward the RGB.Such trends were indeed identified, and were found to be in best correspondencewith a 13.5 Gyr isochrone computed with a parametrization of turbulence usinglogT0 = 6, although a small range of values near this number seemed accept-able. Additionally, the Teff − log g relation suggested a somewhat lower age(see Figure B.1), which was not expected to affect the abundance trends sig-nificantly. The inferred initial lithium abundance in the sample, from an LTEanalysis, was found to be log ε(Li) = 2.54 ± 0.1, in agreement with the bestavailable prediction at the time log ε(Li) = 2.64± 0.03.

We have reexamined the relevant data, and somewhat extended the analysis.In the following sections, we detail our updated analysis.

3.1 Observations

The data used in this analyses were acquired, as per Korn et al. (2007), fromtelescopes in Chile.

Stromgren (uvby, narrow-band) and Johnson (BV I, broad-band) photom-etry was acquired using the Danish 1.54 m telescope at an altitude of 2340meters on the mountain La Silla. Results are given in Table B.1.

Spectroscopy was acquired using the 8.2 meter VLT2 telescope (Kueyen),at an altitude of 2600 meters on the mountain Cerro Paranal. The instrumentFLAMES uses individual fibers for each observed object to feed the light intothe spectrograph UVES, which can use a maximum of 8 inputs at a time, witha maximum resolution of R ∼ 47000.

Seven of the inputs were positioned to observe stars. From an estimateof the exposure times required to reach acceptable levels of signal-to-noise foreach stellar group, an observation scheme was devised. As the SGB group wasunder-sampled as compared to the others, these stars were given precedenceso that their spectra may reach the highest possible quality. To achieve this,two fibers were always dedicated to these stars. The remaining five fiberswere used for the TOP or bRGB stars, with exposure times giving them S/N∼ 50–100 per binned pixel. The last input was positioned on a nearby sky

29

Figure 3.1: Color-magnitude diagram of NGC 6397. The 18 observed starsare marked by crosses. They belong to four groups – from left to right, TOP,SGB, bRGB, and RGB.

area, to provide simultaneous data of the sky background. For faint stars, thesky background constitutes a large portion of the light detected. Hence, theirspectra are corrected by subtracting the sky background spectrum, recordedat the same time. For the RGB stars, all six non-SGB-dedicated fibers wereused for stellar observations, giving no sky correction. At their large apparentbrightness, such corrections are not considered necessary.

UVES was used in the setting Red 580, where the spectra are projected ontotwo CCD sensors, REDL and REDU, which cover the wavelengths λλ4800–5750A and λλ5850–6800 A. Note the small ∼ 100 A gap between CCDs.

3.2 3D and NLTE correctionsFor a differential investigation of the generally small effects we seek, the high-est possible level of realism is required in the spectroscopic analysis. Whererequired and available, we consider abundance corrections as deduced from so-phisticated modelling. We mention here also the treatment considered in theanalysis at additional temperature scales, presented in Chapter 4.

Iron Neutral iron suffers mainly from overionization, causing an underpop-ulation of all levels (Asplund, 2005). This is due to the plentiful iron lines at

30

UV wavelengths, causing ample opportunity for this mechanism to work.The arbitrary factor SH sets the scale of collisional cross-sections. The

resulting collisional coupling between the neutral and singly ionized stage affectsmainly the populations in the neutral species (see e.g. Gehren et al., 2001a,b).

Cram et al. (1980) note that superthermal radiation may result in photonpumping for singly ionized iron, Fe ii. Mashonkina et al. (2010b) however findsthat collisional interactions turn Fe ii to the LTE behavior for any SH > 0.

We use NLTE corrections according to Korn et al. (2003), with SH = 3 inthe analysis of Fe i lines, but examine Fe ii in LTE. The differential analysisof comparing results at different temperature scales is performed in LTE, asonly residual differences are expected from the slight shifts in temperature andsurface gravity.

Titanium We investigate singly ionized titanium, Ti ii, which is the majorityspecies. NLTE effects are therefore expected to be negligible. This is true alsofor 3D effects. Thus, we follow LTE line formation for Ti ii.

Lithium Competing mechanisms dominant under different circumstances arefound by Lind et al. (2009a). Overionization weakens the lines in giants, whileresonance scattering strengthens the line in sufficiently cool stars. Asplund(2005) adds that the increased opacity from resonance scattering in turn feedsadditional line strengthening via decreasing photoionizations, allowing photonsuction to couple recombination of ionized lithium to the neutral ground state.

From the tabulated data of Lind et al. (2009a), we find NLTE correctionsof −0.06 dex for our TOP and SGB stars, roughly −0.01 dex at the bRGB and+0.04 dex for the RGB stars. The downward correction might be an artefactof using 1D atmospheres, as 3D-effects seem to largely cancel this. Sbordoneet al. (2010) performs NLTE calculations in 3D for a grid covering our TOPand SGB stars, showing that NLTE and 3D effects nearly cancel for these stars.We thus expect a slight upward adjustment of the abundances we present here.

The fact that 3D and NLTE effects nearly cancel at the TOP and SGBdoes not imply that the same thing will occur at the giant stages. As thismakes a direct comparison for a theoretical abundance trend between giantsand dwarfs meaningless, we simply exclude the influence of the giant stars onthe comparison, and focus on the TOP–SGB behavior. A proper comparisonof all evolutionary stages may be performed once the full 3D-NLTE behavioris understood.

Magnesium From Gehren et al. (2004), the large photoionization cross-section of magnesium results mainly in overionization. The NLTE correctionsare similar for all evolutionary stages, at between +0.12 and +0.16 dex. Thus,the NLTE trend is similar to the LTE trend. Since such small variations arefound, the same set of corrections (implemented following Gehren et al., 2004)is used for each temperature scale.

Calcium NLTE corrections from Mashonkina et al. (2007) are used. NLTEeffects are found in Korn et al. (2007) to be mainly due to overionization, whichweakens the lines and thus raises the inferred abundances. This is most im-portant in the TOP stars, with decreasing effect toward the lower-temperaturestars. This behavior is due to variations in the continuous opacity at λ < 2028A (i.e. the ground state ionization potential of Ca i), from competing variationsof H− and H+

2 (Mashonkina et al., 2007).

31

Table 3.1: Spectroscopic stellar parameters for group averages.

Group Teff log g ξ [Fe/H](K) (cgs) (km s−1) (dex)

TOP 6254 3.89 2.0 −2.28SGB 5805 3.58 1.75 −2.24

bRGB 5456 3.37 1.73 −2.18RGB 5130 2.56 1.6 −2.12

Corrections are found to alter each line differently, in a manner which flat-tens the trend. Due to the complex behavior of these corrections, we takethe precaution of recomputing them for each effective temperature scale. Theresulting NLTE correction however does not seem to depend on the chosen Teff-scale – it varies by less than 0.01 dex. In other words, the differential analysesat the new temperature scales could just as well have been performed in LTE,as we assumed for magnesium.

Sodium From Gehren et al. (2004), the small photoionization cross-sectionof sodium is found to allow photon suction. Recombined electrons cascadedown to the ground state, which becomes overpopulated. This overpopulationresults in excess absorption, which strengthens the line and shifts the forma-tion depths outwards. We include the element only in differential analyses forstars within the bRGB and RGB respectively, in an effort to identify pollutedcompositions. Thus, we do not expect the NLTE corrections to affect our con-clusions from LTE analyses. To clarify, our results from LTE analyses do showincorrect absolute abundances (low accuracy), but reasonably correct relativeabundances (high precision).

3.3 The original temperature scale

Using the standard iterative process described in Section 2.6, consistent stellarparameters were derived by Korn et al. (2007). Group averages are reproducedin Table 3.1, along with their accompanying elemental abundances in Figure 3.4(or see Table B.4 for a full reproduction of the results).

3.3.1 Anticorrelations and peculiarities

We see some evidence of anticorrelations by plotting the abundances of mag-nesium and lithium against sodium in Figure 3.2. The RGB stars 11093 and14592 both show lower abundances in sodium along with higher abundancesin lithium and magnesium. From Section 1.2, globular clusters are known toconvert magnesium into sodium. These two stars therefore show a less pol-luted composition, in line with the number statistics of Carretta et al. (2009).This should be kept in mind, as their magnesium and lithium abundances areroughly 0.08 dex higher than the group average – corresponding to a strongertrend by roughly one standard deviation for magnesium. Since we do not noteany deviation in the other elements, we do not exclude the four suspectedpollution-affected stars from our analysis, as they improve the quality of theoverall results. Instead, we let an additional data point in the plots for lithiumand magnesium represent the average abundances for these two stars. We referto them in the analysis as the RGB outliers.

32

Figure 3.2: The abundances of magnesium and lithium versus sodium in theRGB sample. Sodium and magnesium, and sodium and lithium both seemanticorrelated, in line with expectations for pollution.

Figure 3.3: Abundances of magnesium versus sodium, and lithium abun-dances as a function of temperature, for the bRGB stars. Top and bottom rowsshow abundances deduced from spectroscopic and photometric b− y tempera-tures, respectively. No anticorrelation is identified. The Lithium abundancesare compared to an arbitrarily shifted 13.5 Gyr T6.0 isochrone, confirming thatthere is indeed a spread in evolutionary state amongst the stars.

33

Figure 3.4: Abundance trends as compared to predictions of traditional stellarmodels (without effects of atomic diffusion – No Diffusion). Triangles representpartial group averages, described in Section 3.3.1. Data is taken from Table B.4.

Strangely, we do not spot these clear anticorrelations for stars in the othergroups. See Figure 3.3 where individual abundances of the bRGB stars aregiven for both spectroscopic and photometric (b − y, following Alonso et al.,1999) temperature scales. The spread in abundances for lithium clearly cor-relates to the temperature, in a manner suggesting simply that the stars arefound at slightly varying stages of evolution. We prove this by including apredicted lithium abundance trend (diffusion model T6.0 at 13.5 Gyr) in thefigure. This correspondence is not an artefact of the spectroscopic tempera-tures, as shown by the photometric results, which despite showing a smallerrange do not change this conclusion. The lack of range in sodium abundancesmay be an artefact of weak lines with amplitudes similar to noise – they shouldtherefore be considered as mere upper limits.

The spread in magnesium abundances coupled with the spectral line behav-ior indicates the bRGB sample consisting of two clumps: stars 3330 and 500949have significantly weaker lines, and correspondingly inferred abundances lowerthan the group average by 0.10 dex. The other three correspondingly lie 0.07dex above the group average.

With such a great range in magnesium abundances (0.17 dex between theclumps, 0.27 dex between the extremes) but no clear pollution signature, it isunclear which values to trust. As such, we add data points for both clumps,and keep in mind that from the perspective of a pollution-affected sample, thehigher abundance should be considered more trustworthy.

For the SGB and TOP stars, the sodium lines are not sufficiently strongfor individual analyses. The magnesium line however is, and shows only slightrandom scatter.

3.3.2 Abundance trendsThe horizontal lines in Figure 3.4 correspond to the prediction of stellar modelsignoring effects of atomic diffusion – any element unaffected by the dredge-up

34

should show a constant surface abundance throughout the stellar evolution.The absolute level of the predicted abundances is shifted to match the obser-vations, corresponding to a shift in the initial abundance.

The discrepancies are obvious. For magnesium, a trend of ∆ log ε(Mg) =0.23 dex between TOP and RGB is seen. The estimated individual differentialerrors of 0.05 dex propagate into an uncertainty of 0.07 dex for the trend, cor-responding to ∼ 3σ. If indeed the two RGB outliers do show the unpollutedabundances, the trend increases to 0.31 dex, at the 4σ level! Such a large abun-dance trend – 0.3 dex corresponds to a factor of 2 – would indeed require propertreatment, due to its importance in e.g. galactic chemical evolution scenarios.For iron, the superior statistics lead the smaller trend of ∆ log ε(Fe) = 0.16 dexto a similar significance at the 3σ level.

For lithium, the comparison of the TOP to the SGB is considered moreimportant than the RGB, as the latter is thought to be affected differentlyby NLTE effects, especially when combined with 3D. As we correct only forNLTE effects, we cannot make a sensible comparison between dwarfs and giants.∆ log ε(Li, SGB−TOP) ≈ 0.12 shows nearly 2σ significance for the abundanceup-turn.

For titanium and calcium, shallow trends of weak significance ∆ log ε ∼ 1σare seen. However, titanium additionally shows the aforementioned indicatorof radiative levitation with a dip at the SGB, at significance 1σ as comparedto the TOP, or 2σ as compared to the RGB.

3.3.3 Comparison to diffusion models

The theoretical isochrones have been computed specifically for Korn et al. (2007,at 13.5 Gyr) and this work (the new models at 11.5 and 12.5 Gyr), by O. Richard(private communication) following Richard et al. (2005).

The simulations are run on a supercomputer, each model using four (outof the 384 available) processors and roughly 1.5 GB of memory. Following theevolution from the pre-main sequence stages up to the turnoff point (corre-sponding to ∼ 10 Gyr) takes roughly 12 hours. The final evolution into theRGB stage (corresponding to ∼ 1 Gyr) requires significantly more computationand possibly even manual intervention. The degree of stratification determinesthe required distribution and number of layers in the model, and the largestpossible timestep able to result in convergence. The process of finding the cor-rect settings is iterative: the wrong parameters will not result in convergence,requiring manual intervention to redefine them. As the abundance distributionchanges, so do the required parameters!

We repeat the comparison of Korn et al. (2007) between the observationalinterpretation and the stellar models with diffusion at an age of 13.5 Gyr, shownin Figure 3.5. Overall, the T6.0 models (logT0 = 6 in (2.1)) fit the observationsbest, matching each trend to within 1σ. The T6.09 model is somewhat flat, withdeviations of ∼ 1σ for magnesium and iron. T5.8 typically deviates by 2σ fromobserved trends. The lithium up-turn at the SGB by nearly 2σ is reproducedonly by the T6.0 model. T6.09 and T5.8 are much too flat and much too strongat ∼ 1.5σ in the respective direction. The case of the RGB outliers indicatesa somewhat lower diffusion efficiency than do the other trends, excluding theT6.09 model, and putting T6.0 at ∼ 1.5σ.

As suggested in Korn et al. (2007), a somewhat lower cluster age than 13.5Gyr is expected. For a cluster age of 12.5 Gyr (Figure B.3), predicted behaviorsare very similar to the 13.5 Gyr models – such a small change in the cluster agedoes not affect the abundance trends much, save for a slight flattening effect

35

Figure 3.5: Comparison of observations to predictions from diffusion modelsfor a cluster age of 13.5 Gyr. Tx.x is shorthand for isochrones using turbu-lent mixing parameter logT0 = x.x. The T6.0 isochrone is found to matchobservations well.

. 0.02 dex. The flattening makes T6.09 now deviate from the iron, magnesium,titanium and lithium trends at more than 1σ. T6.0 still corresponds excellently,at within 1σ to observations for all elements.

At an age of 11.5 Gyr, shown in Figure 3.6, we add the T6.25 model tothe analysis. As before, T6.09 shows a similar but flatter behavior to thehigher ages, again showing weak trends for iron, magnesium, titanium andlithium, off by between 1σ and 1.5σ. T6.25 is much too flat for all but thecalcium trend. Magnesium and iron are much too flat, and titanium does notshow a sufficient downturn at the SGB. For lithium, the TOP–SGB behavior isreversed with a down-turn rather than the observed up-turn (a 3.5σ deviation),as gravitationally settled lithium is mixed into layers where it becomes partiallydestroyed – seen also as a general decrease in the lithium abundance by morethan 0.1 dex.

3.3.4 Comparison to photometryThe results would be strengthened by independent confirmation using othermethods. Notably, photometry may provide this for effective temperatures.From the photometric data (Table B.1), results are presented in Figure 3.7,with full details given in Table B.3.

We use three different calibrations. Alonso et al. (1996, 1999) provide empir-ical calibrations covering the giant and dwarf stages. Casagrande et al. (2010)provides an updated empirical calibration covering only the TOP, and barelythe SGB (the sample is quite thin at log g ∼ 3.6, for natural reasons). Theseempirical calibrations connect photometric observations to accurate ”true” tem-peratures. For each calibration, we use a metallicity of [Fe/H] = −2.0.

Onehag et al. (2009) provides theoretical photometric temperatures, fromintegration of synthetic spectra using model atmospheres similar to those usedin the analysis – MARCS rather than MAFAGS-ODF. This instead can be

36

Figure 3.6: Comparison of observations to predictions from diffusion mod-els for a cluster age of 11.5 Gyr. All models use the same initial elementalabundance – note the large depletion of lithium for the T6.25 isochrone.

thought of as a way of bridging photometry and spectroscopy, by performing aspectroscopic estimate using photometric data. These temperatures should beconsidered consistent within the framework of spectroscopy – an extra way ofchecking that our deduced temperatures are indeed what the observations tellus rather than the product of (sub?)conscious modifications meant to supportour initial suspicions.

At this point, one must choose a philosophy: which temperature scaleshould one feed into the spectroscopic analysis: the true or the consistent one?

The ”true” temperatures found by means of empirical calibration are accu-rately derived from the luminosity-radius-temperature relation. Parallax mea-surements of the distance coupled to estimates of the bolometric luminositygive the effective temperature by its very definition.

On the other hand, the temperatures found by spectroscopic means are bydefinition consistent within a spectroscopic analysis. The synthetic temper-atures are merely an extension of spectrosynthesis to use photometric data.There is however a spectroscopic measure of temperature which is typically ig-nored: the excitation equilibrium. This condition for the effective temperaturerequires that lines of a species at varying excitational energies all indicate thesame abundance. The result for iron lines will typically be lower temperaturesthan those derived using Balmer lines. However, the excitation equilibriumis known not to be trustworthy due to 3D effects. The steeper temperaturegradient of a 3D model atmosphere results in a cooler outer atmosphere, whichaffects the excitational populations in the Boltzmann equation. The temper-ature structure of a 1D model atmosphere results in too low populations inthe low-energy excitational levels, resulting in an overestimate of the requiredabundance for these lines.

Comparing the range of temperatures as deduced by photometry and spec-troscopy, each calibration indicates (by flat or slight upward slopes) that thespectroscopic temperature scale is correct, or that it shows too wide a range of

37

Figure 3.7: Comparison of spectroscopic effective temperatures and all pho-tometric indices for each calibration, as listed in Table B.3. IRFM 2010 refersto Casagrande et al. (2010), IRFM 1999 refers to Alonso et al. (1996, 1999)(dwarf/giant calibrations), and MARCS 2009 refers to Onehag et al. (2009). Aflat curve indicates only an overall absolute shift in temperatures. A negativeslope indicates a wider span of effective temperatures, while a positive slopeindicates a narrower span.

temperatures. Compressing the temperature scale would typically strengthenthe observed abundance trends. Our temperature scale is therefore a conserva-tive estimate.

What about the absolute temperatures at the upper end of the tempera-ture scale? The old set of calibrations indicates lower TOP temperatures thandeduced from spectroscopy. The new IRFM calibration, Casagrande et al.(2010), however suggests roughly 100 K higher TOP temperatures, with quitegood agreement between the three color indices. Two of three indices agree oneven larger increases for the SGB temperatures, but these lie at the very edge ofvalidity for the calibration (see their figure 13). A larger temperature increaseat the SGB than TOP would strengthen e.g. the lithium trend, but again wekeep to a conservative estimate. The fact that this calibration typically indi-cates somewhat higher temperatures than spectroscopic values for metal-poorstars is noted by its authors. Since the calibration does not yet reach giantstars, the implication for the entire temperature scale is unknown. There arehowever indications that the previous results for giants are correct, as thesealready were derived from a large sample of suitable stars.

38

Chapter 4

Results

4.1 Examining different temperature scalesIf, as a qualified guess, we assumed that the RGB temperatures of the originalanalysis were indeed correct, but the TOP temperatures were underestimatedby 100 K as suggested by the new IRFM calibration, this would imply a com-pressed temperature scale. It is not obvious how one would reason when ex-panding it. The simple method is a linear expansion of the entire scale, givingroughly the same relative temperature increase between each stage.

A comparison to the broadening theory employed here, Ali and Griem(1966), is shown in Barklem et al. (2000, Figure 7). ∆Teff(TOP−RGB) differsby only 20 K (Korn et al., 2007), but a peak in the temperature correctionis seen near the SGB. Hence, the broadening theory of Barklem et al. (2000)would expand ∆Teff(TOP − SGB), thus raising the inferred abundances at theTOP. This would for example flatten the abundance trend of lithium somewhat.The precise correction for the temperature of the SGB stars using this broad-ening theory is yet unknown. The sample of Lind et al. (2008) includes starsphotometrically similar to those of our SGB sample. Stars 6760, 13466, 16588,506139, and 502729 in their results show a mean temperature 5834±125 K, i.e.a mere 30 K hotter than our results. Such a small correction is insignificantfor the lithium trend. Thus, we stick to the linear method of expanding thetemperature scale, and leave a detailed analysis to future efforts.

Support for a modified temperature scale is found in Bonifacio et al. (2007),where a temperature increase of 100 K for TOP stars is mentioned, with animplicit intention of keeping the other temperatures fixed. Korn et al. (2007)argues against such a shift on the grounds of its severity: could the errorin temperature between TOP and SGB stars really be 30 %? They furthernote that while such a shift would indeed lessen the trends observed for mostelements, it would not be sufficient to remove the magnesium trend.

If one were to deviate from the spectroscopically consistent temperaturesfound, how would one handle the surface gravities? Spectroscopically, the ion-ization equilibrium condition is applied: the neutral and ionized species of ironshould indicate a consistent abundance. Since we already deviate from thespectroscopically indicated temperatures, we add an additional gravity-scalewhere surface gravities are instead read off from a theoretical isochrone. Thedistinction affects primarily majority species – in our analysis, this means singlyionized iron and titanium, Fe ii and Ti ii.

We thus construct our first temperature scale as a linear expansion by 100 K,with two different sets of log g. As suggested by the original results, we fetchthe isochrone log g values for an age of 12.5 Gyr. In Teff − log g space, the T6.0and T6.09 isochrones are identical, so we need not worry about the free pa-

39

rameter here – turbulent mixing does not affect core helium diffusion, the mainmechanism of atomic diffusion altering the rate of evolution. See Figure B.1 fora comparison of the isochrones at different ages – log g depends only slightlyon age.

As the temperature scale already deviates from the spectroscopic findings,we do not expect the ionization equilibrium condition to fare better. Thus, wechoose the set of isochrone log g values for our primary analysis, and call thisscale +100. The results are not significantly different, so for the sake of clarity,we detail the results using spectroscopic log g (+100e) only in Appendix A.

What would it take to remove the diffusion signatures altogether? FollowingBonifacio et al. (2007), increasing the difference between TOP and SGB stars byroughly 100 K would nearly remove the trends for iron, calcium, and titanium,as well as somewhat flattening the TOP–SGB trend for lithium. To overallflatten the trends, the difference between TOP and RGB stars would need tobe increased by roughly 200 K.

Encompassing both these requirements, we construct yet another temper-ature scale as a linear expansion by 200 K rather than 100 K, resulting in anincrease of ∆Teff(TOP − SGB) by roughly 80 K. Revisiting the photometrictemperatures in Figure 3.7, we find no agreement with such a temperaturescale. It should therefore be considered mainly an experiment in what onemight achieve by investigating the very boundaries of the confidence interval.Curiously, this temperature scale resembles that of Gratton et al. (2001), withTOP stars roughly 220 K hotter, but bRGB stars at roughly the same tem-perature. It is however convincingly shown in Korn et al. (2007) that the highTOP temperatures are probable artefacts of echelle-blaze residuals. Figure 4.1shows representative cases for stars at the lowest S/N of the TOP and SGB.Each spectrum demonstrates that this temperature scale is indeed too hot toreproduce the wings of Hα, with a deviation somewhat larger than 2σ deducedfrom the noise level. This is however dependent on the broadening theoryemployed (we employ Ali and Griem, 1966). Barklem et al. (2000) generatesoverall stronger lines, which in a differential analysis somewhat cancel, thusgiving a similar result.

Yet again, we choose two sets of surface gravities: one in a consistent spec-troscopic manner, and one from comparison to theoretical isochrones in Fig-ure B.1. With the 200 K expansion of the +200 temperature scale, our TOPstars are found at a reasonable evolutionary stage only for the youngest iso-chrone available, at 11.5 Gyr. Note how the ionization equilibrium log g scale(+200e) places the TOP stars below the turnoff point for any isochrone, inconflict with cluster photometry (Figure 3.1). Again, we detail +200 in theprimary analysis, and show +200e only in Appendix A. Our new temperaturescales are presented in Table 4.1, and results are given in full in Table B.5.

Curiously, both isochrone log g scales, +100 and +200, produce Teff-log gconnections consistent with the 13.5 Gyr traditional stellar model, as shown inthe right hand panel of Figure B.1.

To find the corrections to the original results, a reduced analysis is per-formed. For iron, four representative weak lines each are chosen for the neutralspecies, and five for the singly ionized species. For neutral iron, they are chosenat varying excitation energies, ranging from 0.1 to 4.3 eV, to encompass therange of lines’ indicated abundances when the excitation equilibrium conditionis not fulfilled. For the other elements, the full (albeit rather short) line list isused. The full list of lines used in the analysis at the new temperature scalesis presented in Table B.2.

For the temperature scales using log g values from isochrones, the iron abun-

40

Figure 4.1: Comparison of synthetic Hα to observed spectra of stars with thelowest S/N in the TOP and SGB groups respectively. Lines are synthesized atthe original temperature scale (orig; top, red lines), as well as the new scales+100 (middle, green lines) and +200 (bottom, blue lines). While +100 stillreproduces the observed spectral line, +200 is obviously too strong for bothstars.

41

Table 4.1: The new temperature scales, each with two sets of surface gravi-ties. Absolute values, and deviations from the original temperature scale orig(Table 3.1) are given.

Group Teff,avg log gavg ∆Teff ∆ log g+100 – 12.5 Gyr isochrone log g

TOP 6354 4.00 100 0.11SGB 5865 3.73 60 0.15bRGB 5486 3.46 30 0.09RGB 5130 2.56 0 0.0

+100e – Spectroscopic (ionization equilibrium) log gTOP 6354 4.09 100 0.20SGB 5865 3.71 60 0.13bRGB 5486 3.46 30 0.09RGB 5130 2.56 0 0.0

+200 – 11.5 Gyr isochrone log gTOP 6454 4.04 200 0.15SGB 5925 3.73 120 0.15bRGB 5526 3.47 70 0.10RGB 5130 2.56 0 0.0

+200e – Spectroscopic (ionization equilibrium) log gTOP 6454 4.24 200 0.35SGB 5925 3.80 120 0.22bRGB 5526 3.47 70 0.10RGB 5130 2.56 0 0.0

dance is given only by the ionized species, Fe ii. For the temperature scale usingionization equilibrium log g, both ionization stages per definition indicate thesame abundance.

4.2 Examining an expanded temperature scaleHere we investigate the modified abundance trends for the temperature scale+100 (see Table 4.1). ∆Teff(TOP−RGB) has been expanded linearly by 100 K,and log g values are taken from 12.5 Gyr isochrones. See Appendix A for resultswith log g values deduced from the ionization equilibrium of iron (+100e).

4.2.1 Abundance trends

As expected, the trends have flattened somewhat. For magnesium, the trendhas shrunk to ∆ log ε(Mg) = 0.19 dex, significant at 2.7σ – or 3.9σ usingthe least polluted RGB outliers. The overall trend for calcium has flattenedcompletely. For calcium the down-turn on the SGB is no longer significant,and for lithium we find similarly 1.3σ.

The iron trend remains significant at 2.2σ while titanium has indeed flat-tened to insignificant levels, off by just 1.1σ at the SGB down-turn.

Thus, at this temperature scale, the magnesium and iron trends still holdsat large significance. Calcium, titanium and lithium flatten to roughly withinagreement with traditional stellar models.

42

Figure 4.2: Comparison of abundance trends at the temperature scale +100to predictions from diffusion models for a cluster age of 12.5 Gyr.

4.2.2 Comparison to diffusion models

At an age of 13.5 Gyr (Figure B.4), the T6.0 model finds the best agreementwith observed trends. Magnesium, iron and lithium are perfectly reproduced.However, calcium and titanium trends show weak radiative levitation, and de-viate by up to 1.5σ. T6.09 fares slightly worse for calcium and titanium at upto 1.7σ. T5.8 matches the titanium trend nicely, but shows trends too strongin all other cases, disagreeing up to 3σ for iron. The lithium trend however isreproduced at a strongly upward shifted initial abundance. This seems a spu-rious result of the strong slope of the SGB-upturn, which is somewhat offsetin temperature. A slight shift in temperature will dramatically change the in-ferred initial abundance and level of agreement. The magnesium abundance ofthe least polluted RGB outliers is in excellent agreement with the T6.0 model.

At an age of 12.5 Gyr, shown in Figure 4.2, the flattened behavior of T6.0reproduces the trend of each element even better, though still somewhat weakfor titanium at 1.2σ. T6.09 fares somewhat worse for calcium and titanium, offby up to 1.5σ. The magnesium abundance of the RGB outliers shows excellentagreement with T6.0.

At an age of 11.5 Gyr (Figure B.5), T6.09 overall seems better than forthe older isochrones, within the error bars for magnesium, calcium, iron andlithium, but slightly off by 1.3σ for titanium. T6.25 reproduces the calciumand iron trends, but fails similarly for magnesium and titanium at ∼ 1.4σ. Thelithium trend, TOP–SGB, is inverted, off by 2.5σ. The magnesium abundanceof the RGB outliers does not agree with either model.

4.3 Attempting to remove the diffusion signature

Here we investigate the modified abundance trends for the temperature scalelabeled +200 in Table 4.1. See Appendix A for the results with log g valuesdeduced from the ionization equilibrium of iron (+200e).

43

4.3.1 Abundance trends

As expected, the expanded temperature scale causes the trends to overall flat-ten. Magnesium however retains a signature at the 2σ level – or 3σ for the RGBoutliers. Lithium (TOP−SGB) and calcium both flatten to nearly insignificanttrends. Titanium and iron show barely significant ∼ 1.5σ signatures for theTOP–SGB and overall trend respectively.

4.3.2 Comparison to diffusion models

At 13.5 Gyr, results are shown in Figure 4.3. Most importantly, these isochronesdo not reach the alleged TOP temperatures of our stars. As we do wishto compare the behavior anyway, we assume a slight shift in the isochronetemperatures (applied hypothetically – for the models, this corresponds toslightly altered model boundary conditions). The T5.8 models show the cor-rect TOP–SGB behavior for titanium and similarly but too strong for calcium.For lithium and magnesium, trends are too strong by at least 2σ, while ironshows nearly 4σ deviation. T6.0 matches magnesium and lithium, but fails forcalcium and titanium at more than 2σ, and iron at 1.4σ. T6.09 matches mag-nesium and lithium nicely, as well as iron, but fails for calcium and titaniumat > 2σ. The magnesium abundance of the RGB outliers is matched only bythe T6.0 model.

At 12.5 Gyr (Figure B.7), T6.0 and T6.09 match iron, lithium and mag-nesium. The RGB outliers are matched by both sets of models. Calcium andtitanium are similarly off at ∼ 2σ.

At 11.5 Gyr, results are shown in Figure 4.4. T6.25 and T6.09 match ironand magnesium. The RGB outliers match only T6.09. Calcium and titaniumfail similarly at ∼ 2σ for both models. Lithium matches T6.09, but fails at∼ 2σ for T6.25.

4.4 Chromium and nickelAn effort was made to include additional elements in the analysis. For this,NIST was queried for several elements with atomic diffusion abundance trendpredictions available, for any ionization stage. Since metal lines in general areweaker in dwarfs (due to the higher temperature and stronger surface gravity),the spectra of TOP stars decide which lines are possible to detect. For simplicityof identification, the metal-poor field star HD 84937 was used as a proxy forthe TOP stars, due to its available high-quality spectra (see Korn et al., 2003).This way, weak lines were not mistaken for spurious noise features.

Two suitable elements were identified in the spectra: nickel and chromium,both in the neutral ionization stage (Ni i and Cr i). Being iron-like elements,their behavior regarding NLTE effects can be assumed to be iron-like, so linesof the ionized species would have been preferable due to their small NLTE and3D effects (Asplund, 2005).

For chromium, all three Cr i lines of multiplet 7 were detected. Under thephilosophy of quality over quantity, we discard all but λ5206 due to blends withFe i lines.

For nickel, only Ni i λ5476.9 could be detected.Due to NLTE and 3D uncertainties detailed below, we present the results of

our LTE analysis along with theoretical isochrones for possible consideration infuture analyses, but carrying no weight regarding this analysis. Additionally,unlike our other abundance derivations, these are not corrected differentially to

44

Figure 4.3: Comparison of abundance trends at the temperature scale +200to predictions from diffusion models for a cluster age of 13.5 Gyr.

Figure 4.4: Comparison of abundance trends at the temperature scale +200to predictions from diffusion models for a cluster age of 11.5 Gyr.

45

the solar abundance. Therefore, one should expect a displacement in absoluteabundances.

The results of our LTE analysis are presented in full in Table B.4, as wellas along with diffusion model predictions in Figure B.2.

4.4.1 NLTE and 3DChromium A complex interplay of NLTE mechanisms is noted by Berge-mann (2010); Bergemann and Cescutti (2010). Most importantly at low metal-licity, superthermal UV radiation along with insufficient collisional interactionsresult in overionization and photon pumping. The former mechanism connectsto temperature, while the latter connects to metallicity and surface gravity, socorrections are found to be larger for giants than dwarfs. NLTE correctionsare on the order of +0.3 dex for metal-poor dwarfs, but no values are given forgiants.

3D corrections have been identified on the order −0.6 dex for dwarfs and−0.3 dex for giants (Bonifacio et al., 2009). Hence, this difference is on the orderof the discrepancy between our observations and predictions – which notablyshow slopes of reverse signs.

Since NLTE effects are sometimes enhanced and sometimes counteracted by3D effects, it is yet unknown what to expect from 3D-NLTE considerations. Weperform our analysis in LTE, in the hope that corrections from 3D or NLTEanalyses may later be applied, allowing a straightforward comparison to ourresults.

Nickel As another iron-peak element, we expect similar NLTE effects forneutral nickel as for the neutral species of iron and chromium. However, nodetailed calculations seem available for nickel. Consequently, we perform ouranalysis in LTE, in the hope that corrections may become available in thefuture.

If one were to use our results as a benchmark, Figure B.2 shows excellentagreement for the SGB–RGB trend with all available isochrones. The TOPabundance implies either weak turbulent mixing or stronger NLTE effects thanthe other evolutionary stages do. Specifically, the decreased abundance cor-responds qualitatively to overionization, in line with expectations for iron-likeelements.

46

Chapter 5

Conclusions

A lower cluster age results in somewhat flatter trends in the theoretical atomicdiffusion predictions. Flattened trends may in turn be compensated by a some-what lower turbulence efficiency, which allows a greater net effect of gravita-tional settling. However, this relation is inverted for calcium and titanium,where radiative levitation raises their abundances at the TOP. There, weakerturbulence allows radiative levitation to flatten the TOP–RGB behavior. Con-sequently, each element shows a specific dependence on the strength of turbu-lence in both sign and amplitude. In other words, the diffusion signature is notperfectly degenerate as regards age and strength of turbulence.

5.1 The original temperature scale

In our comparisons with observations interpreted at the original temperaturescale, we find that amongst our sample of theoretical isochrones, an age of13.5 Gyr with a turbulence strength of T6.0 gives the best reproduction of theobserved abundance trends. This is in line with the conclusions of Korn et al.(2007). A slightly lower efficiency of turbulence should strengthen and thusreproduce the trends even better.

Our larger sample of theoretical models allows us to deduce that a lowerage, 12.5 Gyr or tentatively even 11.5 Gyr, still falls well within the error bars,especially if the efficiency of turbulence is decreased. The need for weakerturbulence is exacerbated by the raised magnesium abundance of the RGBoutliers.

From the general behavior of the isochrones, by means of linear inter-polation, we find roughly the ±1σ limits for a cluster age of 13.5 Gyr atlogT0 ∈ [6.05, 5.9]. The range is limited upward most strictly by lithium,and downward by calcium and iron. The flat trends of calcium and titaniumdo not contribute to the upper limit. For magnesium, the RGB outliers exac-erbate the need for a low logT0 value, with a 1σ upper limit at logT0 ≈ 5.95,compressing the allowed range for an overall ∼ 1σ correspondence to roughlylogT0 ∈ [6.0, 5.9].

At a cluster age of 11.5 Gyr, the range is shifted somewhat downward, withan estimate of logT0 ∈ [6.0, 5.85], with the upper and lower limits set moststringently by the iron and calcium trends. For magnesium, the RGB outliersset an upper limit of logT0 ≈ 5.95, giving roughly the range logT0 ∈ [5.95, 5.85].

One needs to keep in mind that the effect of the T0 parameter is nonlinear,and most likely does not conform to this simple linear estimate. Our estimateof the overall degenerate behavior of strength of turbulence and cluster age isshown in Figure 5.1.

47

Figure 5.1: The degenerate behavior of age and strength of turbulent mix-ing, from tentative linear extrapolation at the original temperature scale, orig.Filled squares denote the available isochrones. Horizontal bars show roughlythe upper and lower 1σ limits for the overall behavior of the trend. The shadedarea shows the parameter space when considering the magnesium trend of theRGB outliers – an effort to lessen the impact of pollution. The vertical dashedline represents the 11.5 Gyr age deduced from the white dwarf cooling sequence(Hansen et al., 2007), and the horizontal dashed line represents the T6.25 modelpreferred for field stars (Melendez et al., 2010).

Elemental abundance trends at this temperature scale are in tentative agree-ment with the cluster age 11.5 Gyr. Considering Figure B.1, the TOP stars aresome 300 K cooler for the original temperature scale than the turnoff point.This is in disagreement with cluster photometry, which indicates a difference of100–200 K (Korn et al. (2007), F. Grundahl, private communication). Hence,a somewhat higher age of 12.5 Gyr – in 4σ disagreement with the 11.5 Gyrdeduced using the white dwarf cooling sequence, Hansen et al. (2007) – seemsthe best option. Additional isochrones with low efficiency of turbulence at thisage are required to confirm or dismiss this conclusion.

5.2 The extended temperature scaleFrom a diffusion perspective, calcium and titanium still demonstrate slightradiative levitation signatures.

We find the best correspondence for the 12.5 Gyr T6.0 model, Figure 4.2.A somewhat lower turbulence efficiency than logT0 = 6 should give the bestcorrespondence to observed trends, tentatively allowing a shift to 11.5 Gyr.The flat behavior of the iron trend however limits the possibility of a low logT0value.

At 13.5 Gyr, with T6.0 still within the error bars of the iron trend, it shouldbe possible to find a value logT0 ∈ [5.8, 6] giving a rough correspondence to

48

Figure 5.2: The degenerate behavior of age and strength of turbulent mixing,from tentative extrapolation at the +100 temperature scale.

both trends. At this temperature scale, the TOP stars are found just at thetemperature of the turnoff point in the 13.5 Gyr isochrones (in strong conflictwith cluster photometry). As the abundance trends show rapid variation withtemperature at this evolutionary stage, a quite large uncertainty is introducedin our extrapolation of the behavior at lower values of logT0. Thus, the uncer-tainty in these estimates should be considered as greater than for the previoustemperature scale.

At 12.5 Gyr, we find the ±1σ limits logT0 ∈ [6, 5.9], with quite large un-certainty in the lower value. The strictest upper limit is set by the radiativelevitation signature of titanium (a dip at the SGB), and the lower is hintedat by the flat trends of calcium and iron. The magnesium trend of the RGBoutliers does not restrict this range further.

At 11.5 Gyr, linear interpolation sets limits of roughly logT0 ∈ [6.1, 5.85],with considerable uncertainty in both limits. The upper limit is set most strictlyby titanium (at logT0 ≈ 6), and the lower by magnesium. Considering the trendof the RGB outliers, the limits are shifted to logT0 ∈ [6, 5.8]. The resultingrelations are shown in Figure 5.2.

For a cluster age of 11.5 Gyr, the TOP stars are some 200 K cooler thanthe turnoff point (see Figure B.1), in agreement with the cluster photometry.Additionally, elemental abundance trends are found in tentative agreement withthe atomic diffusion models of low efficiency of turbulence, when shifted towardthis age. Hence, the deduced cluster age is in full agreement with Hansen et al.(2007). Additional isochrones with low efficiency of turbulence at this age arerequired to confirm or dismiss this conclusion.

49

Figure 5.3: The deduced initial lithium abundance (dashed line), as foundat the best fitting theoretical isochrones at two different temperature scales,compared to the theoretical prediction from WMAP data (dotted line).

5.3 Attempting to remove the diffusion signatureAt the +200 temperature scale, trends become inconsistent, as shown in Fig-ure 4.3. Calcium and titanium show signs of radiative levitation, while ironand magnesium flatten. The former indicate weak turbulence, while the latterrequire a stronger setting. Additionally, the nearly flattened lithium trend,still hinting at an SGB upturn, indicates a strength somewhere between thetitanium and magnesium requirements.

Thus, no single diffusion model is able to reproduce all trends – differentelements indicate different strengths of turbulence, so the upper and lowerlimits to logT0 do not overlap. The remaining strong trend for the magnesiumabundance at 2σ, as well as the titanium and iron trends at 1.5σ also ruleout traditional stellar models at similar significance as for the diffusion models.This should come as no surprise, as we have already ruled out this temperaturescale on the basis of its disagreement with Hα fitting and photometry. Ratherthan a breakdown of diffusion model predictions, one could regard it as proofthat the free parameter description of turbulence indeed does not allow theprediction of just any arbitrary trends.

5.4 Implications for the primordial lithium abundanceFigure 5.3 shows the deduced primordial lithium abundance of the best modelfor the temperature scales orig and +100. With an uncertainty of 0.06 dexin the predicted primordial abundance, and 0.10 dex in the detected absolutelithium abundance (as adopted from Korn et al., 2007), the comparison has anuncertainty σ ≈ 0.12 dex. This puts the estimated initial lithium abundance atthe original temperature scale (orig) log ε(Li) = 2.48± 0.1 in disagreement byan amount of 70 %, a discrepancy at the 2σ level with theoretical expectationsof log ε(Li) = 2.71± 0.06 (Cyburt et al., 2010). For the expanded temperaturescale (+100), the estimated initial lithium abundance log ε(Li) = 2.55 ± 0.1 isfound in milder disagreement by an amount of 45 %, a discrepancy of roughly1.5σ.

50

The alternative primordial abundance log ε(Li) = 2.66 ± 0.06 (Steigman,2010) shows a 1.5σ discrepancy for the estimated abundance at the originaltemperature scale, and falls just within the 1σ error bars for the expandedtemperature scale.

Altering the age of the cluster seems insignificant for the initial lithiumabundance – for our sample of T6.09 isochrones, a shift of < 0.02 dex Gyr−1 isfound, and the T6.0 isochrones indicate the same initial abundance for ages of12.5 and 13.5 Gyr. However, the efficiency of turbulence is found to stronglydetermine the TOP–SGB behavior. Thus, the correspondence between age andoptimal efficiency of turbulence together give a range of possible initial abun-dances compatible with the spectra investigated. Additionally, the assumedtemperature scale plays a very large role due to the overall shifted lithiumabundances as the temperatures of TOP and SGB stars increase.

It should be noted that +200, the very most extreme scale specifically con-structed to downplay the importance of atomic diffusion, results in the flattesttrends. These are found to be reasonably (although not nearly sufficiently)consistent with the strong turbulence model T6.25, which results in the verylargest amounts of lithium depletion (compare T6.09 to T6.25 in Figure 4.4).The temperature scale constructed as the worst enemy of atomic diffusion turnsout to be its allied.

Following the behavior of Figures 5.1 and 5.2, a decrease in the assumedcluster age couples to a decrease in the required efficiency of turbulence – quiteroughly for +100. As models of low efficiency of turbulence show a strongerlithium abundance trend comparing TOP–SGB, they correspond to a some-what raised initial abundance. Thus, for a cluster age of 11.5 Gyr, we expectsomewhat higher initial lithium abundances than deduced for the available setof isochrones. Hence, the deduced ∼ 1.5σ disagreement between the Big BangNucleosynthesis value from WMAP data, and our observed value at the +100temperature scale should be expected to decrease slightly if suitable theoreticalisochrones are used in the analysis. Additionally, as the effects of 3D and NLTEsomewhat cancel for lithium, one should expect a slight upward correction fromour NLTE abundances.

Considering the Spite plateau, the fact that the lithium surface abundancedepletion shows only weak correspondence to age is important. Otherwise, onecould not reproduce the required flat and thin behavior for a stellar sample withsome variation in age. Additionally, there is some constraint in the efficiencyof turbulence, which must result in a plateau which is flat for stars near theturnoff point. For instance, Korn et al. (2007) note that the weak mixing ofisochrone T5.8 at 13.5 Gyr does not fulfill this constraint. Our results shouldbe compatible with this constraint, as they tentatively suggest a compatiblerange of values somewhere roughly between the models T5.8 and T6.0 at acluster age of 11.5 Gyr for the +100 temperature scale.

51

Chapter 6

Summary

From exploring two expanded temperature scales, at two sets of log g each, weconclude that it is not possible for all observed abundance trends to be ex-plained simply as artefacts of the temperature scale. Flattening the titaniumtrend would require a decrease in ∆Teff(TOP − SGB), which at the same timewould only strengthen the magnesium and lithium trends. Flattening the mag-nesium trend would require expanding ∆Teff(TOP − RGB) by roughly 400 K,in conflict with all temperature indicators, as well as cause the other elementaltrends to steepen yet again. Hence, we expect that the expanded tempera-ture scales +100 and +200 show the overall flattest abundance trends. +100matches the cluster photometry and expected abundance trend predictions forstellar models with atomic diffusion at an age of 11.5 Gyr, in line with the ex-ternal prior of Hansen et al. (2007). An additional set of theoretical isochronesfor this age is required to finalize the analysis.

Regardless of temperature scale, predicted abundance trends match obser-vations only with low efficiency of turbulence. The setting logT0 = 6.25, foundby Melendez et al. (2010) to be in agreement with their sample of metal-poorfield stars, does not reproduce our observations. Notably, such high efficiency ofturbulence causes additional depletion of lithium, which inverts the TOP–SGBtrend, at large significance.

The initial lithium abundance corresponding to model predictions describ-ing our observed abundance trends is expected to be similar to the data inFigure 5.3, discrepant from BBN-WMAP expectations by 2σ for orig, or 1.4σfor +100. The final selection of the optimal theoretical isochrone of +100 mightshift the abundance slightly. 3D-NLTE corrections imply a slight upward shift,of between 0.03 and 0.10 dex (using Sbordone et al., 2010), which puts +100in at least 1σ agreement with BBN-WMAP predictions.

Errors in modelling assumptions could mask shortcomings in the predictivepower of atomic diffusion models. As 3D-NLTE models become available, nodoubt will the results of our NLTE analyses change, albeit slightly. It wouldhowever seem contrived to blame the observed abundance trends, with theirremarkably good correspondence to predictions, on modelling shortcomings.Korn et al. (2007) conclude that as 3D effects are expected to affect elementsquite differently by both sign and magnitude, the abundance trends are notartefacts of 1D modelling.

6.1 Ongoing and suggested future efforts

We have derived effective temperatures using the broadening theory of Ali andGriem (1966). Korn et al. (2007) find that employing Barklem et al. (2000) re-

53

sults in a minor change in the ∆Teff(TOP−RGB). However, ∆Teff(TOP−SGB)should be expected to expand, due to a peak in the temperature correction forthe temperatures of our SGB stars. A large such correction would, for instance,flatten the lithium trend. The precise temperature correction is yet unknown,but expected to be small – see discussion in Section 4.1. The precise effecton this temperature difference and its resulting abundance trends should beinvestigated in future analyses of the present stellar sample.

Lind et al. (2008, 2009b) are textbook examples in (at least) two respectsof how a large-scale study of atomic diffusion should be performed. They useGIRAFFE for investigations covering the evolution from TOP to RGB, foras many elements as possible. Lind et al. (2009b) observed > 300 stars, fora tally of 454 including archive data, sufficient for statistical identification ofunpolluted cluster members. Lind et al. (2008) observe a smaller sample of> 100 stars, but use four wavelength ranges covering suitable lines of fourelements.

Lind et al. (2009b) as well as Gonzalez Hernandez et al. (2009) include mainsequence stars in their investigations. Lind et al. (2009b) find a TOP–SGB up-turn – in line with our results – and a main-sequence behavior suggesting alow efficiency of turbulence. The 3D-NLTE results of Gonzalez Hernandezet al. (2009) however suggest that the main sequence behavior of the theoret-ical models is erroneous, as quantitative agreement is found only for modelsof strong turbulent mixing – specifically, T6.25. This discrepancy should befurther investigated.

The metallicity dependence is not yet understood. The evolutionary depen-dence using globular clusters at varying metallicity need be examined. Cur-rently, the combined results of the globular clusters NGC 6397 and NGC 6752(preliminary results, Korn, 2010) along with the field stars of Melendez et al.(2010) hint at varying optimal strengths of turbulence, with a minimum at[Fe/H] ∼ −2. To this, we can add the meltdown of the Spite plateau for fieldstars at [Fe/H] . −3 (Sbordone et al., 2010).

By narrowing down the range of circumstances where different strengths ofturbulence reproduce the observations, perhaps the physical nature of turbulentmixing will be revealed, in comparison with advanced stellar-structure models(e.g. Talon and Charbonnel, 2004).

The models currently include 28 elements – a list which should be expanded,at the cost of intensified computations. For instance, scandium and barium aredetected in our spectra, but lack theoretical predictions – although scandium isassumed to behave similarly to calcium and titanium. The barium abundancesdetailed in Korn et al. (2007) show no clear trend, with equal values at the TOPand RGB. Such non-trends are of course also a result, and require confirmationin theoretical predictions.

54

Appendix A

Abundance trends using theionization equilibrium

A.1 Examining an expanded temperature scaleFor +100e compared to +100, the iron trend flattens to 1.7σ while the titaniumtrend reaches a slightly larger SGB down-turn ∼ 1.5σ.

As the titanium trend strengthens while the iron trend flattens, they eachrestrict the upper and lower limit respectively to the turbulence efficiency.

Among the 13.5 Gyr models, shown in Figure A.1, T6.09 reproduces theiron trend, while T5.8 reproduces the titanium trend. T6.0 seems the bestcompromise at 1.3σ for iron and 1.9σ for titanium.

At an age of 12.5 Gyr, shown in Figure A.2, both T6.0 and T6.09 reproducethe iron trend well. Titanium however shows 1.5σ for T6.0 and 1.8σ for T6.09.

At an age of 11.5 Gyr (Figure B.6), both T6.25 and T6.09 reproduce theiron trend well. Titanium is off by 1.7σ for both.

A.2 Attempting to remove the diffusion signatureFor the temperature scale +200e, the iron trend flattens completely while thetitanium trend is enhanced to the 2σ level. Thus, two trends – magnesiumand titanium – remain irreconcilable with traditional stellar models at the 2σlevel. Expanding the temperature scale further will exacerbate the problem fortitanium by means of unphysical log g values, and start shifting the currentlyflattened abundance trends toward negative slopes.

No model (Figures A.3, B.8, and B.9) reproduces the titanium trend betterthan ∼ 3σ. For iron, the flat trend matches only the 11.5 Gyr isochrones, withthe best match for T6.25.

55

Figure A.1: Comparison of abundance trends at the temperature scale +100eto predictions from diffusion models for a cluster age of 13.5 Gyr. Other ele-mental trends are identical to Figure B.4.

Figure A.2: Comparison of abundance trends at the temperature scale +100eto predictions from diffusion models for a cluster age of 12.5 Gyr. Other ele-mental trends are identical to Figure 4.2.

Figure A.3: Comparison of abundance trends at the temperature scale +200eto predictions from diffusion models for a cluster age of 13.5 Gyr. Other ele-mental trends are identical to Figure 4.3.

56

Appendix B

Tables and figures

Table B.1: Photometric measurements.

Group Star ID V b− y v − y B − V V − I

TOP . . . . . . . . . . . . 9655 16.200 0.311 0.688 0.396 0.57810197 16.160 0.314 0.695 0.402 0.58512318 16.182 0.312 0.691 0.399 0.581

506120 16.271 0.308 0.683 0.391 0.570507433 16.278 0.308 0.682 0.391 0.570

SGB. . . . . . . . . . . . . 5281 15.839 0.368 0.796 0.476 0.6868298 15.832 0.369 0.799 0.478 0.689

bRGB . . . . . . . . . . . 3330 15.227 0.428 0.932 0.568 0.7846391 15.551 0.412 0.890 0.539 0.760

15105 15.439 0.419 0.909 0.551 0.77023267 15.339 0.424 0.920 0.560 0.778

500949 15.514 0.416 0.900 0.544 0.764RGB . . . . . . . . . . . . 4859 13.815 0.470 1.028 0.636 0.857

7189 13.729 0.474 1.038 0.641 0.86211093 13.551 0.478 1.052 0.651 0.87513092 13.644 0.478 1.047 0.646 0.86814592 13.696 0.476 1.041 0.642 0.865

502074 13.853 0.468 1.024 0.634 0.854

Table B.2: Line list employed for abundance analyses at the newtemperature scales.

Species Line(s) TreatmentFe i λλ5217.3, 5225.5, 5424.0, 6421.3 NLTEFe ii λλ4923.9, 5018.4, 5169.0, 5316.6, 5234.6 LTETi ii λλ5188, 5226 LTECa i λλ6222, 6162, 6439 NLTELi i λ6707 NLTEMg i λ5528 NLTENi i λ5476 LTECr i λ5206 LTE

57

Figure B.1: Comparison of theoretical isochrones at different ages. Datapoints represent the original spectroscopic temperature scale orig, the +100and +200 temperature scales with log g values taken from this very diagram,and the +100e and +200e temperature scales with spectroscopic log g values.log g has not been corrected for helium diffusion (see Section 2.1).

Figure B.2: LTE abundances for chromium and nickel at the original tem-perature scale orig compared to the full set of isochrones. Corrections for thealternative temperature scales are given in Table B.5

58

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2411

0810

8290

610

70..

...

...

.10

6310

55∆(T

OP−

SGB)

.44

943

339

033

344

642

336

952

344

942

8a

Empi

rical

calib

ratio

n(A

lons

oet

al.,

1996

,199

9)b

Empi

rical

calib

ratio

n(C

asag

rand

eet

al.,

2010

)c

The

oret

ical

fit( O

neha

get

al.,

2009

,One

hag

2010

,priv

ate

com

mun

icat

ion)

59

Table

B.4:

Spectroscopicparam

etersand

abundancesfor

individualstarsas

wellasgroup

averages.G

roupaverages

areaverages

oftheeffective

temperature,surface

gravity,microturbulence

andm

etallicity,while

theelem

entalabundancesare

derivedby

applyingthe

groupaverage

stellarparam

etersto

averagedline

spectra.T

heoriginaltem

peraturescale

isused,details

givenin

Section3.3.

Group

StarID

Teff

logg

ξ[Fe/H

]log

ε(Li)log

ε(Mg)

logε(C

a)log

ε(Ti)

logε(N

i)log

ε(Cr)

logε(N

a)

(K)

(cgs)(km

s −1)

NLT

ELT

EN

LTE

NLT

ELT

ELT

ELT

ELT

ET

OP

......9655

62603.85

2.1-2.29

2.245.63

......

......

...10197

62503.95

2.0-2.29

2.245.58

......

......

...12318

62403.90

2.0-2.28

2.155.64

......

......

...506120

62603.85

1.9-2.26

2.325.67

......

......

...507433

62603.90

2.0-2.26

2.325.67

......

......

...SG

B.......

52815800

3.551.7

5-2.25

2.405.75

......

......

...8298

58103.60

1.75-2.23

2.325.75

......

......

...bR

GB

.....3330

54303.35

1.7-2.16

1.275.66

......

......

4.396391

55103.40

1.6-2.14

1.545.93

......

......

4.3915105

54703.40

1.75

-2.181.40

5.88...

......

...4.45

232675370

3.301.8

-2.231.25

5.83...

......

...4.35

5009495500

3.401.8

-2.191.45

5.75...

......

...4.49

RG

B......

48595150

2.651.6

-2.120.90

5.89...

......

...4.16

71895130

2.551.6

-2.140.96

5.81...

......

...4.36

110935100

2.501.7

-2.141.09

5.96...

......

...3.94

130925120

2.551.6

-2.120.89

5.88...

......

...4.25

145925130

2.551.6

-2.121.08

5.96...

......

...4.06

5020745150

2.601.5

-2.101.09

5.75...

......

...4.24

TO

Pavg ....

62543.89

2.0-2.28

2.245.65

4.532.99

3.723.33

...SG

Bavg ....

58053.58

1.75

-2.242.36

5.754.52

2.913.88

3.26...

bRG

Bavg ..

54563.37

1.73

-2.181.38

5.794.55

3.013.91

3.22...

RG

Bavg ....

51302.56

1.6-2.12

0.985.88

4.603.05

3.943.27

...

60

Tab

leB

.5:

Cor

rect

ions

toth

egr

oup

aver

ages

ofTa

ble

B.4

for

the

alte

rnat

ive

tem

pera

ture

scal

esin

trod

uced

inSe

c-tio

n4.

1,an

dan

alyz

edin

Sect

ion

4.2

and

Sect

ion

4.3.

The

empl

oyed

line

list

isgi

ven

inTa

ble

B.2.

Gro

up∆T

eff∆

logg

∆lo

gε(F

e)∆

logε

(Li)

∆lo

gε(M

g)∆

logε

(Ca)

∆lo

gε(T

i)∆

logε

(Ni)

∆lo

gε(C

r)+

100

–12

.5G

yriso

chro

nelo

ggT

OP

100

0.11

0.05

0.07

0.04

0.055

0.075

0.08

0.085

SGB

600.15

0.06

0.045

0.025

0.025

0.075

0.055

0.053

bRG

B30

0.09

0.025

0.025

0.015

0.01

0.035

0.03

0.03

RG

B0

00

00

00

00

+10

0e–

Spec

tros

copi

clo

ggT

OP

100

0.20

0.075

0.07

0.04

0.055

0.10

0.08

0.085

SGB

600.13

0.05

0.045

0.025

0.025

0.07

0.055

0.053

bRG

B30

0.09

0.025

0.025

0.015

0.01

0.035

0.03

0.03

RG

B0

00

00

00

00

+20

0–

11.5

Gyr

isoch

rone

logg

TO

P20

00.15

0.08

0.14

0.08

0.11

0.12

0.16

0.165

SGB

120

0.15

0.073

0.095

0.055

0.065

0.095

0.105

0.11

bRG

B70

0.10

0.047

0.06

0.04

0.045

0.065

0.075

0.08

RG

B0

00

00

00

00

+20

0e–

Spec

tros

copi

clo

ggT

OP

200

0.35

0.145

0.14

0.08

0.11

0.19

0.16

0.165

SGB

120

0.22

0.95

0.095

0.055

0.065

0.13

0.105

0.11

bRG

B70

0.16

0.06

0.06

0.04

0.045

0.085

0.075

0.08

RG

B0

00

00

00

00

61

Figure B.3: Comparison of abundance trends at the original temperaturescale orig to predictions from diffusion models for a cluster age of 12.5 Gyr.

Figure B.4: Comparison of abundance trends at the temperature scale +100to predictions from diffusion models for a cluster age of 13.5 Gyr.

62

Figure B.5: Comparison of abundance trends at the temperature scale +100to predictions from diffusion models for a cluster age of 11.5 Gyr.

Figure B.6: Comparison of abundance trends at the temperature scale +100eto predictions from diffusion models for a cluster age of 11.5 Gyr. Other ele-mental trends are identical to Figure B.5.

63

Figure B.7: Comparison of abundance trends at the temperature scale +200to predictions from diffusion models for a cluster age of 12.5 Gyr.

Figure B.8: Comparison of abundance trends at the temperature scale +200eto predictions from diffusion models for a cluster age of 12.5 Gyr. Other ele-mental trends are identical to Figure B.7.

Figure B.9: Comparison of abundance trends at the temperature scale +200eto predictions from diffusion models for a cluster age of 11.5 Gyr. Other ele-mental trends are identical to Figure 4.4.

64

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68

Glossary

1D The usual stellar atmosphere is calculated using assumptions of a one-dimensional plane-parallel hydrostatic geometry, wherein each layer iscompletely homogeneous. Convection is accounted for by mixing lengththeory, and the free parameters micro- and macroturbulence.

3D Advanced stellar atmospheres calculated with hydrodynamical effects de-scribing convection. Leads to granulation, which changes line formationin a way models are able to reproduce with astonishing accuracy.

Abundance The number of atoms for a certain element X compared to thenumber of hydrogen atoms, log ε ≡ 12+ log(NX/NH) – giving log ε(H) =12.

Atomic diffusion The overall behaviour of several effects causing differentialnet transport of elements: gravitational settling, radiative acceleration,thermal diffusion, diffusion due to concentration gradients (and an ad-hocdescription for turbulent mixing in the models employed here).

Boltzmann equation Describes the atomic populations at different excita-tional levels within one ionization stage, under the assumption of (local)thermodynamic equilibrium.

bound-bound transition Excitation from one energy level to another. Theenergy difference between the levels is fixed, and so the transition occursat a specific wavelength.

bound-free transition Ionization. The electron is given enough energy toleave its atom altogether. Additional energy above the ionization limit isretained as kinetic energy, so the transition does not have an upper energylimit (corresponding to not having a lower wavelength limit). Radiationat any wavelength lower than the ”edge” of a bound-free transition maytrigger it.

bRGB Base of RGB, stars which are just entering the RGB stage of evolution.

convection If a small volume element of higher temperature is set inside alarger volume at lower temperature, the small volume element experiencesbuoyancy. It will tend to rise until this is no longer the case. This causesgranulation.

departure coefficient The ratio of the level populations calculated under theassumptions of NLTE and LTE, bi = nNLTE

i /nLTEi .

dex A relative change given on the scale of logarithms to base 10. See Table B.6for a few examples.

69

differential abundance [X/H] = log ε(X)? − log ε(X), the amount of ele-ment X as compared to the solar abundance..

diffusion signature A differential shift dependent on evolutionary state, e.g.when comparing TOP stars to SGB or RGB stars.

Einstein coefficients Probabilities for transitions between an upper (sub-script u) and lower (subscript l) level: Spontaneous deexcitation (Aul),radiative excitation (Blu) and deexcitation (Bul), and collisional excita-tion (Clu) and deexcitation (Cul).

globular cluster A gravitationally bound system of stars of nearly identicalages (they formed at the same time), which by definition corresponds tothe concept of an isochrone. A comparison between observations of aglobular cluster and an isochrone describing that cluster should give asimple means of evaluating stellar models.

granulation As a small volume element of gas is hotter than its surroundings,it will rise to the surface of the star, where it is suddenly able to radiateaway its surplus of energy. As it grows colder, buoyancy is lost, and itsinks along the edges of the next rising element. This is seen as granula-tion: hotter (brighter) patches surrounded by colder (darker) filaments ina honey-comb or ”foamy” pattern. When observing an entire star at once,thousands of such granules are averaged into the same spectrum. Sincehotter and colder elements result in different line shapes, the averagedline is not reproduced by 1D modeling from first principles.

Hα The most prominent of the hydrogen lines in the visible spectrum, at 6562.8Å. The first (α) in the Balmer-series of lines.

isochrone A set of results from several stellar evolution models with vary-ing mass parameters but otherwise identical in e.g. metallicity and (bydefinition) age.

Line formation Transitions between energy levels of an atom may emit (spon-taneously or stimulated by another photon) or absorb a photon at awavelength corresponding to the transition energy gap. The absorptionmechanism drains photons from the specific wavelengths of the relevanttransitions, forming an absorption line. By matching a synthetic spec-trum to what is observed, the stellar parameters are deduced spectroscop-ically, within the framework of a model atmosphere and some appropriateline formation mechanisms.

LTE Local thermodynamical equilibrium. An adaptation of thermodynamicequilibrium to stellar atmospheres, where all processes are said to belocal. The radiation intensity is set by the Planck function, and theatomic populations are set by the Saha and Boltzmann equations. Thegeneral case is known as NLTE.

macroturbulence A fudge-factor representing large-scale convective motion.It is applied to a synthetic spectrum as a convolution (often Gaussian),which smears spectral lines without affecting their strength.

metal Any element which is neither hydrogen nor helium, i.e. has an atomicnumber Z > 2, or a mass number A > 5.

70

metallicity Typically given as [Fe/H], the amount of iron as compared to thesun in logarithmic units. Defined this way since iron lines dominate thetypical visual spectrum, and iron with some exceptions seems to representthe overall amount of metals well.

microturbulence A fudge-factor required in 1D atmospheres to synthesizesaturated lines. Turbulence here refers only to small-scale motion, i.e. alocal Maxwell-Boltzmann velocity distribution.

model atmosphere A set of useful parameters given at the range of depthsinvolved in line formation: e.g. temperature T , gas pressure Pg, electronpressure Pe, density ρ.

NLTE Calculations of atomic level behaviour without the assumption of localthermodynamical equilibrium (LTE) – hence non-LTE, NLTE. Atomicpopulations depend on effects of the (non-local) radiation field, which inturn depends on the atomic populations.

NLTE correction Given as the difference between the more complex analysis(NLTE) and the simpler (LTE) analysis, e.g. ∆ = log εNLTE − log εLTE.A similar concept exists for 3D modeling.

optical depth Given as a dimensionless number τ describing the exponentialremoval of light along the line of sight. Larger numbers refer to greaterdepths, smaller numbers to the outer regions of the photosphere. Opticaldepth relates to the effective temperature as Teff = T (τ = 2/3) in theso-called grey case.

Planck function Describes the spectrum and luminosity of a black-body, im-plying (local) thermodynamical equilibrium.

population The relative amount of atoms at a certain ionization stage orexcitational level, as compared to the total amount of atoms for thatelement.

restricted NLTE NLTE calculations are performed only in the line forma-tion, without feedback effects on the model atmosphere.

RGB Red giant branch, where stars past the main sequence end up after thecore hydrogen has been depleted.

RGB outliers As detailed in Section 3.3.1, the sample of RGB stars shows ananticorrelation trend typical of pollution within globular clusters, relatingabundances of magnesium and lithium to sodium. Two out of the six RGBstars – the outliers – seem to suffer less from this effect, and should thusbe the preferred abundance indicator for the affected elements.

S/N Signal-to-noise ratio.

Saha equation Describes the atomic populations at different ionization stagesunder the assumption of (local) thermodynamical equilibrium.

SGB Sub-giant branch, stars just past the TOP.

species An element in a specific ionization stage.

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Logarithmic increase/decrease (dex) 0.01 0.05 0.10 0.30 0.50Linear increase (%) 2.3 12 26 100 216Linear decrease (%) −2.3 −11 −21 −50 −68

Table B.6: A few typical values on the logarithmic abundance scale.

temperature scale The adopted set of consistent effective temperatures in ananalysis, which are usually possible to shift without affecting the resultsof the analyses. This because spectroscopic mechanisms may give onlydifferentially consistent rather than absolute values.

the cosmological lithium-problem A disagreement between the calculatedprimordial lithium abundance from standard big bang nucleosynthesiscalibrated by WMAP data, and the abundance inferred from the so-calledSpite plateau in metal-poor stars. The disagreement is on the order of0.3− 0.5 dex, corresponding to a factor 2− 3.

TOP Turnoff-point, where a dwarf has nearly exhausted the available hydro-gen in the core, and rapidly transforms into a red giant across the SGBand bRGB stages over a billion years (cf. roughly ten billion years on themain sequence).

traditional stellar models Stellar models which do not treat the effects ofatomic diffusion.

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Nomenclature

+100 A new temperature scale. Effective temperatures are arbitrarily shifted,surface gravities are taken from a 12.5 Gyr isochrone. Parameters aregiven in Table 4.1.

+100e Another temperature scale, following +100 but with surface gravitiesspectroscopically determined using the ionization equilibrium – thus e forequilibrium. Parameters are given in Table 4.1.

+200 Another temperature scale, parameters are given in Table 4.1. Surfacegravities are taken from a 11.5 Gyr isochrone.

+200e Another temperature scale, following +200 but with surface gravitiesspectroscopically determined using the ionization equilibrium – thus e forequilibrium. Parameters are given in Table 4.1.

Iν Intensity.

I Ionization potential.

Jν Mean intensity.

N Total number of particles, of specified type.

Sν Source function.

Teff Effective temperature. The representative temperature of a stellar photo-sphere, defined from the relation to luminosity and radius in the Stefan-Boltzmann law, L = 4πR2σT 4

eff.

Xν Parameter X depends on the frequency.

[X/H] Differential abundance of element X.

log ε(X) Absolute number abundance of element X, log ε ≡ 12+ log (NX/NH).

α Line extinction coefficient (i.e. opacity).

χ Excitational energy.

log gf Oscillator strength and degeneracy. These parameters determine rela-tive line strengths within a species.

log Logarithm to base 10. See also dex.

log g Surface gravity of a star.

The sun. Indicates solar values when used as subscript.

τν optical depth, describes the exponential reduction of intensity through amedium.

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ξ Microturbulence parameter.

f Oscillator strength.

g Degeneracy of the atomic level.

jν Line mean intensity.

n Number density of particles, of specified type.

orig The original temperature scale, as deduced spectroscopically by Kornet al. (2007). Parameters are given in Table 3.1.

X ii The singly ionized stage of element X.

X i The neutral stage of element X.

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