Atomic Diffusion in Old Stars

FYSAST Examensarbete 15 hp November 2010

Testing parameter degeneracies

Thomas Nordlander

Institutionen för fysik och astronomi Department of Physics and Astronomy

Abstract Atomic Diffusion in Old Stars

Thomas Nordlander

Teknisk- naturvetenskaplig fakultet UTH-enheten The predicted primordial lithium abundance differs from observations of unevolved halo stars on the Spite plateau by a factor two to three. Surface depletion due to Besöksadress: atomic diffusion has been suggested as a cause of this so-called cosmological lithium Ångströmlaboratoriet Lägerhyddsvägen 1 problem. Evolutionary abundance trends indicative of atomic diffusion have previously Hus 4, Plan 0 been identified in the metal-poor globular cluster NGC 6397 ([Fe/H] = -2), with stellar parameters deduced spectroscopically in a self-consistent manner. Abundances Postadress: of five elements (Li, Mg, Ca, Ti, and Fe) were found to be in agreement with stellar Box 536 751 21 Uppsala structure models including the effects of atomic diffusion and a free-parameter description of turbulent mixing at the lowest efficiency compatible with the flatness of Telefon: the Spite plateau. 018 – 471 30 03

Telefax: It is our aim to evaluate the interplay of modelling assumptions and theoretical 018 – 471 30 00 predictions under various priors, e.g. the independent age determination using the white dwarf cooling sequence, and the high efficiency of turbulent mixing recently Hemsida: found compatible with halo field stars. http://www.teknat.uu.se/student We perform self-consistent spectroscopic abundance analyses at an expanded effective temperature scale inspired by results of new photometric calibrations from the infrared flux method. The resulting abundances are compared to predictions in a grid of theoretical isochrones, chosen in light of the priors for age and efficiency of turbulent mixing. We find that the observed abundance trends are not artefacts of the effective temperature scale, as it cannot be arbitrarily modified to flatten all trends. The inferred abundance trends seem to be in agreement with predictions for an age compatible with the white dwarf cooling sequence, and a limited range of weak turbulent mixing. The inferred initial lithium abundance of these stars is merely 30 % lower than the primordial abundance, discrepant at 1.5 standard deviations. Hence, a stellar solution to the cosmological lithium problem is still within reach.

Handledare: Andreas Korn Ämnesgranskare: Bengt Edvardsson Examinator: Bengt Edvardsson UPTEC FRIST08 000

Sammanfattning

Vid Big Bang sammanfogades protoner (som utgör kärnan i väteatomer) och neutroner 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 specifik modell 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 i tiden man går, desto lägre halt av metaller bör man därför finna. Mycket me- tallfattiga stjärnor (en metallhalt som är en hundradel av solens) förväntas därför vara mycket gamla, och hos dessa finner man i undersökningar sedan 1980-talet praktiskt taget alltid samma halt litium (ungefär en litiumatom per fem miljarder väteatomer). Eftersom de observerade stjärnorna visar variation i yttemperatur och massa är det långt ifrån självklart vad för slags process som skulle kunna ändra litiumhalten precis lika mycket för alla dessa stjärnor, så att litiumhalten antingen ökas eller sänks från värdet vid Big Bang till det observerade. Det hela blev än mer problematiskt tjugo år senare, när satelliten WMAP lämnat data som ihop med modeller förutspådde en litiumhalt två–tre gånger högre än den observerade. Oförmågan att förklara denna skillnad kallas det kosmologiska litiumproblemet. Modeller som beskriver stjärnors utveckling bygger på en rad fysikaliska antaganden, och vissa förenklingar. En av dessa förenklingar är att man igno- rerar diffusionsprocesser, där olika ämnen genom sina mikroskopiska rörelser tillåts göra en netto-vandring genom stjärnan med hjälp av någon fysikalisk mekanism. Gravitationen får alla ämnen tyngre än väte att sjunka mot botten (liksom kaffepulver tenderar att sjunka till botten). En gradient (övergång) i koncentrationerna tenderar att utjämnas av atomernas slumpvandring (en droppe 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 de halter 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 att motsvara 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 stjärnans ytlager minskar. Ju äldre stjärnan är, desto längre tid har den haft på sig att låta ämnen sjunka inåt. När kärnreaktioner i stjärnans centrum förbrukat den tillgängliga vätgasen börjar dess atmosfär expandera, så att den över en miljard år blir till en röd jätte. Under denna omvandling sätter konvektion – storskalig omblandning – in i atmosfären, så att de ämnen som tidigare sjunkit blandas upp i ytlagren igen. Eftersom endast stjärnans ytlager är observerbara måste man därför under- söka likartade stjärnor både innan och efter att atmosfären utfört denna omblandning. Skillnaden i ämneshalter utgör en trend, och påvisar hur stor mängd som dolts i stjärnans djupare lager. Svårigheten ligger i hur man uppskattar ämneshalterna. Den vanliga metoden kallas spektroskopi, där man delar upp ljuset 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 en viss energi, och varje sådan energi motsvarar en viss våglängd hos ljus – fotoner av viss energi. Atomövergången interagerar med fotoner av rätt våglängd, så att ljus vid just den gällande våglängden saknas i stjärnans spektrum – detta kallas en spektrallinje. Mängden saknat ljus, spektrallinjens styrka, svarar mot ämneshalten samt den kvantmekaniska sannolikheten för att övergången skall ske. 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 undersökta stjärnor. I den artikel varpå denna undersökning bygger användes den spektroskopiska metoden konsekvent. Man undersökte 18 stjärnor i en metallfattig stjärnhop på 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 de spektroskopiskt uppskattade yttemperaturerna är något för låga. Denna studie bygger därför på samspelet mellan de uppskattade yttemperaturerna och vilka ämneshalter och trender de leder till. Vi har även undersökt hur dessa passar teoretiska förutsägelser då olika kraftig turbulens och olika ålder hos stjärnorna antagits. 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 ihop med teoretiska förutsägelser för en stjärnhop vid 11,5 miljarder års ålder. Detta resultat 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 uppsättning teoretiska modeller inte var tillräckligt väl vald – just där resultaten verkar passa bäst har vi ingen modell att jämföra med. Vår härledda ursprungliga litiumhalt avviker med ungefär 45 % från den förutspådda halten (istället för de klassiska 100 − 200 %), en avvikelse av viss statistisk betydelse. Denna skillnad förväntas minska något – kanske ända till 25 % – när man tar hänsyn till mer avancerade modeller för linjebildning. De slutgiltiga detaljerna beror på just vilken teoretisk modell som visar sig passa resultaten bäst, men det handlar inte om några omvälvande skillnader. Contents

1 Introduction 9 1.1 The cosmological lithium problem ...... 10 1.2 Globular clusters ...... 12

2 Theory 13 2.1 Stellar models with atomic diﬀusion ...... 13 2.2 Stellar model atmospheres ...... 16 2.3 Spectral line formation ...... 17 2.3.1 The strength of a spectral line ...... 17 2.3.2 Spectral line broadening ...... 18 2.3.3 The application of spectral lines ...... 18 2.4 Relaxing modelling assumptions – 3D and NLTE ...... 19 2.4.1 Hydrodynamical models – 3D ...... 20 2.4.2 NLTE ...... 20 2.4.3 NLTE mechanisms ...... 22 2.5 Spectroscopic observations ...... 23 2.6 Method ...... 24

3 Background 29 3.1 Observations ...... 29 3.2 3D and NLTE corrections ...... 30 3.3 The original temperature scale ...... 32 3.3.1 Anticorrelations and peculiarities ...... 32 3.3.2 Abundance trends ...... 34 3.3.3 Comparison to diﬀusion models ...... 35 3.3.4 Comparison to photometry ...... 36

4 Results 39 4.1 Examining different temperature scales ...... 39 4.2 Examining an expanded temperature scale ...... 42 4.2.1 Abundance trends ...... 42 4.2.2 Comparison to diffusion models ...... 43 4.3 Attempting to remove the diffusion signature ...... 43 4.3.1 Abundance trends ...... 44 4.3.2 Comparison to diffusion models ...... 44 4.4 Chromium and nickel ...... 44 4.4.1 NLTE and 3D ...... 46

5 Conclusions 47 5.1 The original temperature scale ...... 47 5.2 The extended temperature scale ...... 48 5.3 Attempting to remove the diﬀusion signature ...... 50 5.4 Implications for the primordial lithium abundance ...... 50 6 Summary 53 6.1 Ongoing and suggested future eﬀorts ...... 53

A Abundance trends using the ionization equilibrium 55 A.1 Examining an expanded temperature scale ...... 55 A.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 and Spite (1982) find a roughly constant lithium abundance among warm, metal- poor (old) halo stars. From the view of galactic chemical evolution where metallicity increases with time, this should straightforwardly correspond to the primordial lithium abundance, and this was the favored interpretation for roughly twenty years. Calculations of Big Bang Nucleosynthesis coupled with WMAP cosmological data now suggest a primordial lithium abundance a factor of two to three greater. Thus, the cosmological lithium problem: observations strongly disagree with predictions, suggesting a serious error in standard cosmology, fundamental physics, or stellar physics. However, long before the cosmological lithium problem was realized, Michaud et al. (1984) found that the effects of atomic diffusion, traditionally neglected in stellar models, should correspond to a factor two or more depletion from the formation abundance of these stars. Thus, this mechanism could explain the discrepancy. Korn et al. (2007) confirm this finding in an investigation of 18 stars at varying evolutionary state (between the turnoff point and the red giant branch) in the metal-poor globular cluster NGC 6397. The observational signatures of atomic diffusion are detected at statistical significance up to 3σ in five elements; magnesium, calcium, titanium, iron, and lithium itself – shown in comparison to theoretical predictions in Figure 1.1. From the comparison of theoretical predictions to observations, an initial lithium abundance of log ε(Li) = 2.54 0.1 is found to agree within the 1σ error bars with the then predicted primordial abundance log ε(Li) = 2.64 0.03 – the logarithmic 0.1 dex difference corresponds to just 25 %, significantly less than the cosmological lithium problem. However, new data from WMAP and nuclear physics raise the primordial abundance to log ε(Li) = 2.71 0.06. Additionally, the stellar abundances cannot 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 %, a disagreement at the 2σ confidence level. Thus, it seemed appropriate to reexamine the previous results in the light of new findings. For instance, updated photometric calibrations suggest higher effective temperatures for the turnoff-point stars than those derived spectroscopically. Higher effective temperatures weaken lines of minority species, e.g. the investigated lithium line, leading to a greater abundance. However, changing the assumed effective temperature would affect all observed elements, requiring a proper reexamination of the effects. Additionally, the precise effects of atomic diffusion depend on the strength

9 Figure 1.1: Comparison of observations to predictions of diffusion models for a cluster age of 13.5 Gyr. Tx.x is shorthand for isochrones using turbulent mixing parameter log T0 = x.x. The T6.0 isochrone is found to match observations well at the temperature scale introduced by Korn et al. (2007). Here, abundance trends are incompatible with traditional stellar models, which indicate a constant abundance for each element except lithium, where dilution effects set in at the SGB. of turbulent mixing, for which only a free-parameter description exists. The process is not yet properly understood, so within the framework of the current description of turbulent mixing, one must find the range of the free parameter in concordance with observations. This work extends the investigation of Korn et al. (2007) by examining the abundances on two new temperature scales: one suggested by photometry, and one designed to remove the abundance trends. We attempt to extend the analysis with additional elements, but due to modelling shortcomings cannot draw conclusions from their behavior. Finally, we attempt to determine whether an external determination of the cluster age (Hansen et al., 2007) is compatible with the abundance trends, and reexamine the implications for the cosmological lithium problem. The cosmological lithium problem, galactic chemical evolution, and the ap- plications and shortcomings of globular clusters are reviewed in Section 1.1 and Section 1.2. The process of atomic diffusion, stellar modelling, and the method of spectroscopy are detailed in Chapter 2. The investigation of Korn et al. (2007) is reexamined in Chapter 3, and our new investigation is detailed in Chapter 4. The results are evaluated in Chapter 5.

1.1 The cosmological lithium problem

With a primordial mix of only the hydrogen, helium, and lithium created by Big Bang Nucleosynthesis a few minutes after the Big Bang, generations of stars have enriched the gas content of galaxies later on. The gas content of a galaxy is known as the interstellar medium, and its content of elements heavier

10 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 evolve into the giant stages, they also form metals in the core, mainly ranging from carbon to iron, with details depending mainly on the stellar mass. Due to the characteristics of the thermonuclear reactions, the stellar mass also strongly controls the rate of evolution. Lifetimes range from roughly ten billion years for stars similar to the Sun, to just millions for stars a hundred times more massive. Stars less massive than the Sun are expected to have lifetimes longer than the age of the Universe. As the stellar core depletes the available hydrogen, the nuclear reaction rates decrease, and the decreasing core luminosity causes it to contract. By conser- vation of entropy, the stellar atmosphere expands, causing the star to leave the dwarf state on the main sequence and turn into a giant. As the atmosphere expands, convection causes mixing in a zone reaching increasingly deeper. As layers containing processed core material are reached, the first dredge-up occurs when these elements mix into observable layers. Later, stellar winds expel parts of the outer atmosphere, with whatever elements may be present. Addi- tionally, a sufficiently massive star will experience the supernova process, where it rapidly creates and destroys metals in great quantities, and expels at least a large fraction of them. This is the process of chemical enrichment. As the nuclear processes are limited to the stellar core, surface metallicity could be expected to stay unaffected during the star’s lifetime, at least until it turns toward the giant stage. Its surface metallicity should match that of the material from which the star formed. Assuming rapid mixture processes resulting in a homogeneous interstellar medium, one would imagine that stars with surface compositions of lower metallicity formed from an interstellar medium of low metallicity, indicating a high age for the star. As a rough indicator for a star’s age, this does not seem inconsistent with the idea of short lifetimes for massive stars – observations of the most metal-poor stars do indicate low masses 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 known as the Spite plateau: the lithium abundance barely changes with stellar parameters (Spite and Spite, 1982). The na¨ıve interpretation tells us that this must be 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 of Big Bang Nucleosynthesis were coupled with the cosmological data gathered by WMAP – specifically the factor ηB ≡ nB/nγ, i.e. the number ratio of baryons to photons. The results were a primordial lithium abundance on the logarithmic scale of log ε(Li) ≈ 2.64 0.03, which compares to the observed Spite 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 exacerbated the problem by raising the predicted primordial lithium abundance to log ε(Li) = 2.71 0.06 – an increase by ∼ 20 %(Cyburt et al., 2010). Steigman (2010) includes additional observational data in the modelling. The standard Big Bang Nucleosynthesis model coupled with observed deuterium abundances, and the non-standard model with additional equivalent neutrino flavors, 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 abundance slightly dependent on mass (Meléndez et al., 2010) or metallicity (Ryan

11 et al., 1999; Sbordone et al., 2010). Obviously, something must be wrong with our understanding of either fundamental physics, cosmology, or galactic or stellar evolution. Atomic diffusion allows lithium to settle toward depths where it is heavily depleted, as suggested by Michaud et al. (1984) and identified observationally as a solution to the problem by Korn et al. (2006b, 2007) – upon which this work is based. It involves measurements of stars at different evolutionary stages in a globular cluster (see Section 1.2), with comparisons to predictions of stellar models incorporating the relevant mechanisms. The theory involved is reviewed in Section 2.1, and we revisit the results concerning lithium in Section 5.4.

1.2 Globular clusters

Globular clusters are dense collections of stars formed within a galaxy, with nearly the same age and composition, as well as the same distance to us. In this picture, they are known as simple stellar populations – globular cluster stars at varying evolutionary stages should differ only in their initial mass, which determines the other attributes of the star. A more massive star will typically 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 that of the galaxy at the time, which roughly indicates their age. However, the cluster stars are expected to form over an extended period, which allows the very most massive and short-lived stars to enrich (pollute) their surroundings before all cluster stars have formed. The intracluster gas (corresponding to the interstellar gas of galaxies) from which the stars form can be thought to mix sufficiently rapidly that the composition of a specific star depends strictly on when it formed, as compared to the complete stellar formation history of the cluster – similar to the na¨ıve view of stellar formation in a galaxy. The cluster thus contains both first and second-generation stars, formed from either a pristine or polluted composition. The process of pollution can be thought of as a small-scale version of galactic chemical evolution. Rather than completely shaping the metal content of the second generation, mere details are shifted, due to the limited time available. Additionally, following Gratton et al. (2004), one could expect stars of the first generation to during their lifetime accrete polluted material from companions or nearby stars. In the accretion scenario, one would expect the signature of pollution to disappear from the surface as the star evolves into stages with deep convection. In the formation scenario, this would not occur, as the star has a homogeneously polluted composition. For instance, AGB stars experience hot bottom burning where the core reaches convective layers, causing oxygen to be depleted in the CNO-cycle, and magnesium and neon respectively to deplete as they form aluminum and sodium via proton-capture. These relations are seen as anticorrelations for O-Na and Mg-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 – for NGC 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 intracluster medium once pollution has commenced, while less massive clusters risk instead having it ejected before pollution takes place. Additionally, less massive clusters – like NGC 6397 – can be expected to contain fewer very massive stars, giving a lower overall expected effect of pollution.

12 Chapter 2

Theory

2.1 Stellar models with atomic diﬀusion

Traditional stellar structure models solve coupled diﬀerential equations. A simpliﬁed 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 within is transfered outward.

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

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

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

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

• Thermal diﬀusion.

• Net movement due to concentration gradients.

• Turbulent mixing, counteracting concentration gradients by means of additional 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. The turbulent mixing coefficient is given by Richard et al. (2005) as

[ ]− ρ 3 DT = 400DHe(T0) (2.1) ρ(T0) where T0 is the free parameter, represented as a reference temperature, determining the efficiency of turbulence. DHe(T0) is the helium atomic diffusion coefficient at T0. The factor 400 arbitrarily determines the overall strength,

13 while the exponent −3 determines the localization of the mixing, as calibrated by observations of the solar beryllium abundance being similar to the initial abundance. Lithium destruction occurs at layers where T ∼ 2.5 · 106 K (or 6 log T ∼ 6.4). By setting T0 < 2.5 · 10 K, turbulent mixing is restricted to the outer layers, where temperatures are sufficiently low that depletion is min- imized. In general, atomic diffusion causes heavy elements to diffuse toward greater depth. Details depend on specific properties of each element, when radiative levitation competes with gravitational settling. At the surface layers, this is seen as a general depletion of metals. For low-mass, metal-poor stars, a thin convection zone exists in the outermost regions, observable as granulation. Set- tled elements rest below this region. As the star reaches the giant stages, the convection zone expands inward until the layers with enhanced metallicity are reached, causing the settled elements to reemerge. Later on, this includes elements 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 toward the core causes an offset in the hydrogen abundance. This in turn causes nuclear reactions to more rapidly exhaust the core’s hydrogen, giving the star a more rapid evolution into the giant stages. An accelerated evolution for any given star translates into lower masses for the the stars of an isochrone, which in turn results in a shift toward lower effective temperatures. Hence, the turnoff point will be found at a somewhat lower temperature (Richard et al., 2002). Additionally, the settling of helium from the surface layers causes a change in 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 heavy elements. Lessening the effects of atomic diffusion, a more efficient turbulence (higher reference temperature T0) causes flattened diffusion trends. As turbulence 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. There are also exceptions to the general behavior of turbulence in the outer regions. With a sufficiently high efficiency of turbulence, settling is shifted toward the deep, hot layers where lithium is destroyed. With a non-destructive efficiency of turbulence, lithium is simply deposited within. As the convective zone expands inward, these layers are reached, and lithium resurfaces, giving an up-turn when comparing the SGB to the TOP. This causes an enhancement in lithium surface abundances at the subgiant stage. As convection runs deeper, it reaches layers depleted by thermonuclear reactions, causing the lithium surface abundance to drop according to the traditional effect of dilution, just before the first dredge-up. With a higher efficiency of turbulence, the deepening convection zone finds only depleted layers, instead causing the surface lithium abundance to drop smoothly, without the up-turn. For other elements, the efficiency of turbulence mainly sets the depth of settling, which determines when the deepening convection zone causes resurfacing of deposited material, and how much is settled. For calcium and titanium, radiative levitation may dominate over gravitational settling, unless suppressed by turbulent mixing. With a low efficiency of turbulent mixing, these elements behave akin to lithium at a high efficiency of turbulent mixing, with a slight down-turn at the subgiant stage as compared to the turnoff point. This is due to radiative levitation hindering gravitational settling near the surface. Further into the star, the higher temperature results in

14 ionization, which reduces the cross-section of the atom. Thus, these elements are levitated near the surface, retaining the initial abundance, while deeper layers experience gravitational settling, thus depleting. As the convection zone reaches these layers, dilution results in the lithium-like down-turn. With a high efficiency turbulence, the effect of radiative levitation is somewhat canceled. The mass of a star determines its rate of evolution, and during the post-main sequence evolution, the convective envelope grows. Additionally, the mass of a star determines its structure. Notably, the temperature gradient determines whether a layer is stable or affected by convection. The connection is that a lower mass results in a thicker convective envelope. Combining these facts, we find the thinnest convection zones for stars near the turnoff point – these are the 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 the turnoff point. Additionally, a lower age leaves less time for atomic diffusion to work, 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-trivial behavior from the connection between age (mass) and luminosity. Add to this the 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 cooler stars at a given evolutionary stage.

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

• Stronger turbulent mixing causes ﬂatter abundance trends when comparing evolutionary stages – the trends end up mostly similar to non-diﬀusive stellar 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 turbulent mixing. A slight up-turn in lithium abundance at the SGB in a weak mixing scenario, or extra depletion giving a down-turn in a high eﬃciency turbulence scenario.

• Possibly ﬂatter trends for a set of younger stars.

In light of this, a comparison to observations should be performed for a set of similar stars at varying evolutionary stages, for instance ranging from unevolved stars to subgiants and red giants with the eﬀects of dredge-up visible. Abundances of both lithium and other elements should be analyzed, as different behaviors are expected from the same set of parameters. A suﬃciently large such comparison should constrain both upper and lower limits to the free parameter of turbulent mixing, or dismiss it altogether if these limits do not overlap.

15 2.2 Stellar model atmospheres

Spectroscopic observations of real stars give a connection to stellar models by means of line synthesis. This involves calculating a synthetic spectrum under some 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 relation between temperature T and a depth parameter – e.g. optical depth τ. For 3D descriptions, a single geometrical depth x simultaneously holds warm upﬂows and cool downdrafts, and thus does not correspond to a single relation for T and the depth parameter. The computation of a model stellar atmosphere follows a similar recipe as traditional models of stellar structure, described in the previous section. In our description, following Rutten (2003), we let subscripts i and j refer to any two atomic levels, while subscripts u and l refer to the speciﬁc upper and lower levels of a transition. Under the LTE approximation, all processes are said to be local and in thermodynamical equilibrium locally, so ionization and excitation populations are given directly by solving the Saha and Boltzmann equations. The non-LTE case is described in Section 2.4.2. The Boltzmann equation describes the ratio of any two excitational level populations, [ ] n g χ − χ i = i exp − i j (2.2) nj gj kT where excitational level i has population ni, statistical weight gi, and excitational energy χi relative to the ground level. The transition energy is χu − χl ≡ hc/λ, with λ the transition wavelength. The Saha equation describes the ratio of populations in successive ionization stages: ( ) [ ] 3/2 Nr+1 1 2Ur+1 2πmekT − I = 2 exp (2.3) Nr Ne Ur h kT where ionization stage r has population Nr, partition function Ur, and ionization potential I. Ne is the number density of electrons, and me the electron mass. Combining these equations for all ionization stages of an element trivially gives the ratio of the population for an ionization stage to the total population of the element. Combined, the Saha and Boltzmann equations describe the population of any excitational level at any ionization stage for an element. Supposing that all partition functions and statistical weights are known (which they generally are – many tabulations are available), a given temperature instantly solves for the precise population distributions. This is the strength and point of LTE analyses. See Section 2.4.2 for an overview of alternatives. The radiative transfer equation is given as (following Gray, 2005)

dIν = Sν − Iν (2.4) dτν where Iν is the intensity, and τν the optical depth. Averaging the intensity over all space angles results in the mean intensity, ∫ 1 J ≡ I dΩ. (2.5) ν 4π ν The source function depends on emission and absorption coeﬃcients. These have contributions from both the continuum and from individual spectral lines.

16 Under the diﬀusion approximation and LTE, Jν = Sν = Bν. Solving the radiative transfer equation (2.4) including all its constituents, and the equation of hydrostatic equilibrium, results in the distribution of temperature and partial gas 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 level populations. Again following Rutten (2003), superscript l and c refer to vari- ables describing either the line or the continuum, and subscripts retain their previous 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 constant is 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 excitation) are also possible, with probability constants Bul and Blu respectively. They are related as Bul = Blugl/gu, via the statistical weight g of each level.

• Collisional interactions have probability constants Cul and Clu.

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

j 2hc 1 Sl = ν = (2.6) ν 2 gunl αν λ − 1 glnu 2hc 1 = [ ] ≡ B (T ) (2.7) λ2 hc − ν exp λkT 1 with, the statistical weights g and number densities n. jν and αν are the line emission and extinction coeﬃcients. While (2.6) is the general result, (2.7) is true only in LTE by the assumption (2.2). Its result is known as the Planck function, which describes the spectral distribution of black-body radiation. Line opacity is given by the line extinction coeﬃcient. In LTE,

hc αl = (n B − n B ). (2.8) ν 4πλ l lu u ul

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

αl R = ν (2.9) κν

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

17 2.3.2 Spectral line broadening The shape of a spectral line at a specific depth is described by a Voigt function, which convolves Gaussian and Lorentzian (dispersion) profiles. For a sufficiently strong line, the Gaussian profile dominates the core while the Lorentzian profile dominates the wings. Each is due to different processes, which broaden (damp) lines. The damping factor of a Lorentz profile is γ 1 ψ(ν − ν0) = ( ) (2.10) 4π2 − 2 γ 2 (ν ν0) + 4π where ν is the investigated frequency, ν0 the (resonant) frequency of the transition, 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 energy given by the Heisenberg relation. Its shape is a Lorentzian profile, with a damping constant given by the spontaneous decay probabilities (i.e. the inverse lifetime) 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, and is important in hot stars. The latter dominates in cool stars, where collisions with neutral atoms alter energy levels. Both have Lorentzian profiles. Resonance broadening and the linear Stark effect are relevant for hydrogen atoms, by interactions with other hydrogen atoms or charged particles. While resonance broadening too has a Lorentzian profile, linear Stark broadening is more complex (but becomes important only for hot stars). Doppler broadening results from the velocity distribution of particles, giving varying Doppler shifted energies. The usual Maxwell-Boltzmann velocity distribution gives a Gaussian profile. Turbulent broadening is for simplicity separated into microscopic and large-scale components, microturbulence and macroturbulence, respectively. The effect of rotation is the same as for macroturbulence.

2.3.3 The application of spectral lines The final shape of the spectral line is determined by its full distribution of formation depths, where the varying local parameters determine the interplay of the source function and line opacity, as well as the different broadening components. Each observed wavelength of the spectral line has a different set of formation depths, as the line itself determines the optical depth. This set is known as the contribution function. In the core of a line, enough light is blocked that one cannot see very far into the atmosphere – the contribution function is thus dominated by the cooler outer regions of the atmosphere. In the wings, less light is blocked, so the contribution function centers on deeper layers. Importantly, different parts of the line have different contribution functions. For a sufficiently strong line, the contribution function is shifted into the outer atmosphere where model stellar atmospheres become less realistic (typically, they lack a chromosphere). Such lines cannot be completely described by the stellar model, limiting their usefulness instead to the wings. The equivalent width is a measure of the total line strength. The equivalent width of a spectral line is given by the width of a hypothetical, perfectly rectangular, 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 the majority species. Thus, minority species are temperature sensitive, as is the continuous opacity. Altering the gravity changes the electron pressure Pe, i.e. Ne in our form of the Saha equation (2.3), again shifting the ionization fraction. For minority species, the effect on line opacity cancels that of the continuous opacity (mainly contributed by H−). Majority species however find only an altered continuous opacity. Thus, only majority species are pressure sensitive. These mechanisms give a means of evaluating the effective temperature and surface gravity of a star. If lines of different excitational energy show different abundances, the excitational populations in the atmosphere do not match those of the model – thus, the wrong temperature (distribution) is being used. For lines from different ionization stages of an element, incompatible abundances indicate an incorrect surface gravity. These conditions are known respectively as the excitation and ionization equilibrium. The effect of altered abundances is given by the curve of growth. Three typical regions exist, • Weak lines, where the Doppler core dominates. The equivalent width and line depth vary linearly with the number of absorbers (ε). • Saturation, where the central depth asymptotically reaches the maximum value. • Strong lines, where the broadening causes significant optical depth in the line wings. For cool stars, the pressure dependence of van der Waals broadening determines the behavior of the wings. For a saturated line, broadening by microturbulence shifts part of the line into the wings, thus strengthening the line by avoiding the saturated wavelength region. For a weak line, only broadening is gained, as the line core is not saturated. The weak-line behavior of microturbulence is akin to the effect of macroturbulence and rotation, which act only as external broadening profiles. The Doppler shift of large-scale regions moving in different directions merely result in a ”smeared” line, which is broadened but not strengthened. Thus, the equivalent width is hardly affected. It seems a good idea to include the additional information contained in line profiles rather than blindly computing and comparing equivalent widths. The comparison of observed and synthesized line profiles can be performed manually, with comparison by eye. Lind et al. (2008, Figure 2) instead performs an automated analysis, where the line shape is considered in a manner somewhat akin to the photometric method of measuring flux within a specified wavelength region. One could also imagine a setup where observed line shapes are evaluated and corrected for spurious noise features and blending lines manually, before an automated fitting procedure begins.

2.4 Relaxing modelling assumptions – 3D and NLTE

We give here a summary of the mechanisms and ideas involved in more advanced modelling. A proper review is given by Asplund (2005), and mechanisms are detailed in Rutten (2003). Understanding of these mechanisms is important as they could possibly undermine any study not taking care to correct for their eﬀects.

19 2.4.1 Hydrodynamical models – 3D 3D models use physical descriptions of hydrodynamics to describe the processes in a stellar atmosphere, which results in granulation. Simply put, granulation describes an observable process of convection: warm upflows and cool downdrafts each have different temperature structures, giving components of several different line shapes, averaged into a shape which does not naturally arise from 1D atmospheres. Instead, the free parameters micro- and macroturbulence are ad-hoc varied to reproduce the spectrum. Inherent asymmetries of these mo- tions result in a slight line asymmetry which is not reproduced by free parameters in 1D models. With warm upflows and cool downdrafts, the temperature T and depth parameter, e.g. optical depth τ, at a single geometrical depth x is inhomogeneous. The radiative transfer equation (2.4) must be solved including these horizontal inhomogeneities. 3D models are thus, by their nature, quite computationally heavy. An important result of 3D models is found for low-metallicity stars, where the low metallicity causes lower opacity. This decreases the effect of line heating (line blanketing), which leads to lower temperatures in the outer atmospheric regions. Heat transfer is then mainly adiabatic, rather than radiative. The resulting temperature structure is thus steeper than in corresponding 1D atmospheres. The cool outer layers in the steep temperature structure result in larger populations for neutral minority species, making such lines stronger than in 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 sophistication in NLTE analyses (described below) does not allow such simpliﬁcations. One solution is to improve the line opacity treatment in the 3D model, resulting in highly realistic but challenging computations. Alternatively, a 1D stellar atmosphere where the temperature distribution is gathered from the averaged results of a 3D simulation can be used in existing NLTE codes. Such an atmosphere is called 1.5D, as it somewhat retains the superior 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 for photons and collisionally dominated transitions. When this does not hold, reality can no longer be approximated by LTE – we must consider non-local effects: non-LTE (NLTE). These arise when radiative interactions dominate over collisions. For cool stars, metals rather than hydrogen act as the main electron donors. At sufficiently low metallicities, the decreasing electron pressure allows radiative processes 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,

dn ∑N ∑N 0 = i = n P − n P (2.11) dt j ji i ij j=6 i j=6 i where zero equality represents the assumption of steady state, ni is the population of level i, the transition rates are Pij = Aij + BijJ¯ν + Cij, with the

20 Einstein coefficients Aij for radiative emission, Bij radiative absorption (stimulated emission) and Cij collisional excitation (deexcitation), and J¯ν is the mean intensity over all angles (2.5) as averaged for the transition. The true populations are described by their resulting departure coefficients: NLTE ni bi = LTE . (2.12) ni These can be put into the NLTE form of the line source function, modifying (2.7). For bound-bound transitions, 2hc 1 b Sl = [ ] ≈ u B (2.13) ν 2 b hc ν λ l exp − 1 bl bu λkT meaning that the result is a modified Planck function: if the lower level population suffers larger depletion than the upper level, the resulting source function will be stronger than the Planck function. This is known as superthermal radiation. If however both level populations show similar departures, bu ≈ bl, then l ≈ Sν 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, ( [ ]) l hc LTE − bu − hc αν = blnl Blu 1 exp , (2.14) 4πλ bl kT λ 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 source function (2.13), which depends on the populations from (2.11) rather than the Saha and Boltzmann equations’ resulting Planck function (2.7). Thus, the intensity depends on the populations, which depend on the intensity. Such analyses cannot be restricted to a single transition, all transitions of a single ionization stage of an element, or even all transitions of all all ionization stages of an element, as transitions of other elements may still also influence these equations at the relevant wavelengths. Analyses where this is assumed nonetheless are known as restricted NLTE, described further below. Additional complexity is found with time dependent considerations, which remove the zero-variation of the statistical populations (2.11), of importance for hydrodynamical 3D stellar atmospheres. An important note is that the levels and transitions required for the NLTE analysis are not only determined by the lines used in the spectroscopic abundance determination. A sufficiently large difference in a few lines might result in a species-wide effect. Importantly, the lower ionization stage needs a more or less complete description of the energy levels up to near the ionization potential - although closely spaced excitation levels may be combined into superlevels to lessen the computational cost. This is important since a collisional coupling is required between the continuum electron reservoir and the highest excitation levels to describe collisional ionization and recombination properly. The gap between the highest excitational levels and the ionization potential should be on the order of the typical kinetic energy, as given by a Maxwell-Boltzmann velocity distribution (Mashonkina et al., 2010a). If the highest energy levels are not sufficiently well described, the impact of collisional 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. This is applied to the line formation routines as a correction from the LTE populations, giving the correct line shape during the spectroscopic part of the analysis.

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

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

Restricted NLTE The usual treatment of NLTE does not allow the deviant level populations to feedback on the model atmosphere – the stellar atmosphere is still calculated under the assumption of LTE, while line formation follows NLTE. This works under the assumption that one is dealing with a trace element, whose deviation from the LTE assumption will not heavily impact on the surroundings. For instance, an ample electron donor would aﬀect the overall electron 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 mechanisms Overionization In the deeper, hotter layers of the stellar atmosphere, the high temperatures cause the Planck function to shift toward shorter wavelengths, with a rapid increase in the UV for cool stars. This steepness of the Planck function creates a non-locality in the radiation field: the much greater UV flux from deeper layers is able to seep into the colder layers. This is known as superthermal radiation, since Jν > Bν. If a species has transitions with bound-free transitions at UV energies, the superthermal UV radiation will cause greater photoionization than expected from LTE. Photoionization is thus the main mechanism for overionization. Overionization amounts to a decreased line opacity, making the lines weaker, which corresponds to an underestimated abundance and thus a positive abundance correction. The species affected (for cool stars: the neutral stage of the element) will then suffer a depletion in those energy levels, which if numerous will drain all populations of the species. This effect is important for minority species, where a relatively small variation in the populations of the majority species translates into a large relative change. Therefore, since line opacities are sensitive to populations, they should be sensitive to a variation in the amount of ionization. It is also enhanced by steeper temperature structures, since they connect to a steeper variation in the Planck function. Use of a model atmosphere with insufficient steepness in the temperature will thus result in NLTE masking (Rutten and Kostik, 1982). Importantly, the steeper temperature structures of 3D atmospheres will result in stronger overionization effects (Asplund, 2005).

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

Photon pumping Similar to overionization, but instead for excitational transitions. Transitions with energies in the UV region affected by superthermal radiation will suffer excessive excitation, causing an overpopulated upper level. As the lower population depletes into the upper level, the source function increases, causing a weakened line. The increase in the affected upper level also affects its other transitions, according to (2.13). Again, this validates the claim that the line list employed in a NLTE analysis needs to be more or less complete. The weakened line corresponds to an underestimation of the abundance.

Photon suction If a highly excited energy level, near the ionization potential, is connected via a chain of radiative transitions all the way down to near the ground state, the so-called photon suction mechanism may become important. An atom in the ground state of the ionized stage could momentarily recombine with a free electron. With a suﬃciently large transition probability compared to the photoionization probability, this mechanism could step in before the atom is reionized, instead causing the electron to lose its excess energy as a series of emitted photons. This results in under-ionization, where the 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 line itself causes local depletion of radiation, Sν < Bν. This correction corresponds directly to a strengthened line, i.e. again an overestimation of the abundance.

2.5 Spectroscopic observations

Converting light as gathered by a telescope into a spectrum requires a disperser, known as a spectrograph. It uses a diﬀraction grating, composed of angled rulings (grooves) covered in a thin reﬂective layer, causing light to disperse by means of wavefunction interference. The grating equation describes the distribution of interference maxima in the dispersed light, (Gray, 2005, Chapter 3)

nλ = θ = sin α + sin β (2.15) d 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, d the spacing of rulings, and n the integer order of the (symmetrical) interference pattern counting from n = 0 at the central maximum. For a pointlike source composed of multiple wavelengths, the small relative shift in wavelength for a specific interference order n will give a small shift of β, resulting in adjacent images – a spectrum. For larger values of n – higher orders – the dispersion becomes greater, corresponding to higher resolution. Gratings working at large n – on the order of 100 – are known as échelles. Interference causes the formation of multiple maxima, but it is diffraction which determines their amplitude, following a sinusoidal which diminishes with increasing 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 a stronger signal. Introducing a phase shift will affect only the diffractive behavior, while retaining the interference pattern. Thus, the diffraction envelope is shifted so that the largest amplitudes go to the desired large value of n for some intended wavelength. The phase shift is set by the angle of the rulings in the grating, and the method is called blazing. With a blazed spectrum, the output is a single long line, which is projected onto a photosensitive chip - a CCD. The higher the resolution, the longer this line will become, thus making the observable bandwidth limited by the chip size. However, this uses only a single line of a chip, which would need to be unreasonably wide. The solution involves cross-dispersing the spectrum using additional 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 be compensated. Failing to do so results in a ”wavy” continuum, which is difficult to compensate for in strong lines, of particular interest for effective temperature determination using broad Balmer lines like Hα. Having attempted to removed the blaze from each order, they are merged into a single continuous spectrum. As the orders overlap somewhat, there are no ”holes” in the spectrum. There may however remain blaze residuals from the individual orders. The wavelength range of an order n which does not overlap the adjacent orders n 1 is known as the free spectral range. If a wide line is not positioned within the free spectral range of any single order, its behavior can be affected more by artefacts of order merging than by the stellar parameters one wishes to investigate. Examining faint stars, each exposure lasts roughly one hour. Often, several such 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, each spectrum must be individually shifted to match a set of well-behaved and easily identifiable lines. This compensates for the varying radial velocity giving a Doppler shift – introduced by e.g. stellar motion, and Earth’s rotation and motion – which would otherwise introduce duplicate or smeared lines. The resolving power R ≡ λ/∆λ is chosen to conform to both observational constraints and physical requirements. R must be set so that ∆λ becomes sufficiently small that adjacent lines be resolved (individual lines are typically not fully resolved), while at the same time giving a reasonably large S/N for the resulting spectrum to be useful. The former constraint is important in crowded line 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 the profile of the hydrogen Balmer lines, primarily Hα. Due to stellar atmospheres being dominated by hydrogen, this line is very strong. This means that the core forms in the outer layers above the photosphere, where our models no longer apply. Instead, one focuses on the wings, which do form in the photosphere. The atomic physics describing hydrogen is however quite difficult, and much research has been devoted to it. For instance, Barklem et al. (2000) introduces hydrogen self-broadening terms from quantum mechanical calculations. These are however not applied in this analysis of Hα, which instead follows the resonance 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 the broadening theories. In lieu of complete 3D hydrodynamical atmospheres, the 1D hydrostatic atmospheres commonly used apply so-called fudge-factors to simulate hydrodynamical effects: the microturbulence and macroturbulence. The microturbulence affects the strength of saturated lines, while weak lines are merely broadened. Extending a weak line analysis to include strong lines, the correct microturbulence is found when lines show an abundance independent of line strength. Macroturbulence and rotation are applied as an external profile, often a Gaussian convolution which smears the lines without affecting their strength. When fitting single lines in an abundance analysis, only the elemental abundance and external profile are varied, until the synthetic line matches the spectrum in both depth and overall shape. The abundance found to match a single spectral line is not an absolute measure. Due to uncertainties in the atomic data (primarily the log gf value), different lines will generally indicate different abundances. This problem is addressed by performing a line-by-line differential analysis: The same line is investigated in the solar spectrum, using a model with the standard solar parameters and identical atomic parameters. Dividing the stellar abundance by the solar abundance derived for that line then gives the differential abundance, [ ] ε(X)?,i [X/H]i = log . (2.16) ε(X) ,i If different atomic parameters are used, the line-by-line differential analyses is simply set to involve the log gf value, by comparing the solar measurement log gfε ,i to the stellar measurement log gfε?,i. Taking the mean of the abundances for each line, we find the stellar abundance of the element, with some random scatter. Supposing that the physics of the star is sufficiently similar to the Sun, the systematic errors introduced by incorrect atomic parameters will by design cancel in a differential analysis. An alternative to this type of line-by-line differential analysis, is an inverted analysis of solar lines, described by e.g. King et al. (1998). Setting the best known solar element abundance, the atomic parameters are adjusted until the model successfully recreates the solar spectrum. This cancels out the same systematic effects as above. Either way, there will still be other systematic errors. For instance, the comparison of a metal-poor star to the Sun is deficient in the simple fact that the Sun is not a metal-poor star. Lines sufficiently strong to be detected in a metal-poor star will be so strong in the Sun as to be affected by different formation mechanisms. We attempt to avoid this by including the weakest lines possible. The method of finding the stellar parameters uses an iterative process of the above concepts.

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

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

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

25 4. Adding lines of singly ionized iron, i.e. the majority species, the surface gravity is varied until the same abundance is found for both species. At this point, the stellar parameters have been determined. If the result at this point is found to be inconsistent with the Balmer lines, step 1 and forth should be reiterated until convergence is reached.

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

For certain elements, the available lines may be too weak for single star measurements – 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. Using the mean stellar parameters of each group, the group-averaged spectra of each line might gain the S/N required for a precise abundance determination. The resulting elemental abundances then carry a number of diﬀerent possible errors.

• Model discrepancies, e.g. employing an insuﬃcient broadening theory for eﬀective 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, resulting in an oﬀset normalization. This leads to lines being identiﬁed as being either too strong or too weak. For the elemental abundances, this type√ of error is alleviated by taking the average of several spectra, causing a N reduction of the noise.

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

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

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

Having identiﬁed the errors, their magnitude must be estimated.

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

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

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

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

In lieu of proper statistics for our small samples of metal lines, we assume typical differential uncertainties of 0.05 dex for each element in each group. In Section 3.3 we mention the cases where star-to-star scatter is found to be larger than this, and is thought to be real. Using our above estimated errors, the significance of an observed diffusion 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 convective envelope on the subgiant-giant branch causes a different behavior. The typical estimated errors of√ 0.05 dex, at both the TOP and RGB, propagate into 2 2 a standard deviation σ = σ1 + σ2 = 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)

Chapter 3

Background

We give here a short overview of the original investigation, published as Korn et 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 stellar models including atomic diffusion suggest a general settling of elements at the turnoff point, which gradually resurface as the stars evolve toward the RGB. Such trends were indeed identified, and were found to be in best correspondence with a 13.5 Gyr isochrone computed with a parametrization of turbulence using log T0 = 6, although a small range of values near this number seemed acceptable. Additionally, the Teff − log g relation suggested a somewhat lower age (see Figure B.1), which was not expected to affect the abundance trends significantly. The inferred initial lithium abundance in the sample, from an LTE analysis, was found to be log ε(Li) = 2.54 0.1, in agreement with the best available 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), from telescopes in Chile. Strömgren (uvby, narrow-band) and Johnson (BVI, broad-band) photometry was acquired using the Danish 1.54 m telescope at an altitude of 2340 meters 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 instrument FLAMES uses individual fibers for each observed object to feed the light into the spectrograph UVES, which can use a maximum of 8 inputs at a time, with a maximum resolution of R ∼ 47000. Seven of the inputs were positioned to observe stars. From an estimate of the exposure times required to reach acceptable levels of signal-to-noise for each stellar group, an observation scheme was devised. As the SGB group was under-sampled as compared to the others, these stars were given precedence so that their spectra may reach the highest possible quality. To achieve this, two fibers were always dedicated to these stars. The remaining five fibers were 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 stars are 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, the sky background constitutes a large portion of the light detected. Hence, their spectra are corrected by subtracting the sky background spectrum, recorded at the same time. For the RGB stars, all six non-SGB-dedicated ﬁbers were used for stellar observations, giving no sky correction. At their large apparent brightness, such corrections are not considered necessary. UVES was used in the setting Red 580, where the spectra are projected onto two CCD sensors, REDL and REDU, which cover the wavelengths λλ4800–5750 A˚ and λλ5850–6800 A.˚ Note the small ∼ 100 A˚ gap between CCDs.

3.2 3D and NLTE corrections

For a diﬀerential investigation of the generally small eﬀects we seek, the highest possible level of realism is required in the spectroscopic analysis. Where required and available, we consider abundance corrections as deduced from sophisticated modelling. We mention here also the treatment considered in the analysis 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 affects mainly 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 photon pumping for singly ionized iron, Fe ii. Mashonkina et al. (2010b) however finds that 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 in the analysis of Fe i lines, but examine Fe ii in LTE. The differential analysis of comparing results at different temperature scales is performed in LTE, as only residual differences are expected from the slight shifts in temperature and surface gravity.

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

Lithium Competing mechanisms dominant under different circumstances are found by Lind et al. (2009a). Overionization weakens the lines in giants, while resonance scattering strengthens the line in sufficiently cool stars. Asplund (2005) adds that the increased opacity from resonance scattering in turn feeds additional line strengthening via decreasing photoionizations, allowing photon suction to couple recombination of ionized lithium to the neutral ground state. From the tabulated data of Lind et al. (2009a), we find NLTE corrections of −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 artefact of using 1D atmospheres, as 3D-effects seem to largely cancel this. Sbordone et al. (2010) performs NLTE calculations in 3D for a grid covering our TOP and 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 SGB does not imply that the same thing will occur at the giant stages. As this makes a direct comparison for a theoretical abundance trend between giants and dwarfs meaningless, we simply exclude the influence of the giant stars on the comparison, and focus on the TOP–SGB behavior. A proper comparison of all evolutionary stages may be performed once the full 3D-NLTE behavior is understood.

Magnesium From Gehren et al. (2004), the large photoionization cross- section of magnesium results mainly in overionization. The NLTE corrections are 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 are found, 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. NLTE eﬀects are found in Korn et al. (2007) to be mainly due to overionization, which weakens the lines and thus raises the inferred abundances. This is most important in the TOP stars, with decreasing eﬀect toward the lower-temperature stars. This behavior is due to variations in the continuous opacity at λ < 2028 A˚ (i.e. the ground state ionization potential of Ca i), from competing variations − + of H and H2 (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 −2.28 SGB 5805 3.58 1.75 −2.24 bRGB 5456 3.37 1.73 −2.18 RGB 5130 2.56 1.6 −2.12

Corrections are found to alter each line differently, in a manner which flattens the trend. Due to the complex behavior of these corrections, we take the precaution of recomputing them for each effective temperature scale. The resulting 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 analyses at 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-section of sodium is found to allow photon suction. Recombined electrons cascade down to the ground state, which becomes overpopulated. This overpopulation results in excess absorption, which strengthens the line and shifts the formation depths outwards. We include the element only in differential analyses for stars within the bRGB and RGB respectively, in an effort to identify polluted compositions. Thus, we do not expect the NLTE corrections to affect our conclusions from LTE analyses. To clarify, our results from LTE analyses do show incorrect absolute abundances (low accuracy), but reasonably correct relative abundances (high precision).

3.3 The original temperature scale

Using the standard iterative process described in Section 2.6, consistent stellar parameters were derived by Korn et al. (2007). Group averages are reproduced in 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 magnesium and lithium against sodium in Figure 3.2. The RGB stars 11093 and 14592 both show lower abundances in sodium along with higher abundances in lithium and magnesium. From Section 1.2, globular clusters are known to convert magnesium into sodium. These two stars therefore show a less polluted composition, in line with the number statistics of Carretta et al. (2009). This should be kept in mind, as their magnesium and lithium abundances are roughly 0.08 dex higher than the group average – corresponding to a stronger trend by roughly one standard deviation for magnesium. Since we do not note any deviation in the other elements, we do not exclude the four suspected pollution-aﬀected stars from our analysis, as they improve the quality of the overall results. Instead, we let an additional data point in the plots for lithium and magnesium represent the average abundances for these two stars. We refer to them in the analysis as the RGB outliers.

32 Figure 3.2: The abundances of magnesium and lithium versus sodium in the RGB sample. Sodium and magnesium, and sodium and lithium both seem anticorrelated, in line with expectations for pollution.

Figure 3.3: Abundances of magnesium versus sodium, and lithium abundances as a function of temperature, for the bRGB stars. Top and bottom rows show abundances deduced from spectroscopic and photometric b − y temperatures, respectively. No anticorrelation is identiﬁed. The Lithium abundances are compared to an arbitrarily shifted 13.5 Gyr T6.0 isochrone, conﬁrming that there is indeed a spread in evolutionary state amongst the stars.

33 Figure 3.4: Abundance trends as compared to predictions of traditional stellar models (without effects of atomic diffusion – No Diffusion). Triangles represent partial 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 other groups. See Figure 3.3 where individual abundances of the bRGB stars are given 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 are found at slightly varying stages of evolution. We prove this by including a predicted lithium abundance trend (diffusion model T6.0 at 13.5 Gyr) in the figure. This correspondence is not an artefact of the spectroscopic temperatures, as shown by the photometric results, which despite showing a smaller range do not change this conclusion. The lack of range in sodium abundances may be an artefact of weak lines with amplitudes similar to noise – they should therefore be considered as mere upper limits. The spread in magnesium abundances coupled with the spectral line behavior indicates the bRGB sample consisting of two clumps: stars 3330 and 500949 have significantly weaker lines, and correspondingly inferred abundances lower than the group average by 0.10 dex. The other three correspondingly lie 0.07 dex above the group average. With such a great range in magnesium abundances (0.17 dex between the clumps, 0.27 dex between the extremes) but no clear pollution signature, it is unclear 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, the higher abundance should be considered more trustworthy. For the SGB and TOP stars, the sodium lines are not sufficiently strong for individual analyses. The magnesium line however is, and shows only slight random scatter.

3.3.2 Abundance trends The horizontal lines in Figure 3.4 correspond to the prediction of stellar models ignoring 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 observations, 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 differential errors of 0.05 dex propagate into an uncertainty of 0.07 dex for the trend, corresponding to ∼ 3σ. If indeed the two RGB outliers do show the unpolluted abundances, the trend increases to 0.31 dex, at the 4σ level! Such a large abundance trend – 0.3 dex corresponds to a factor of 2 – would indeed require proper treatment, 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 dex to a similar significance at the 3σ level. For lithium, the comparison of the TOP to the SGB is considered more important than the RGB, as the latter is thought to be affected differently by NLTE effects, especially when combined with 3D. As we correct only for NLTE effects, we cannot make a sensible comparison between dwarfs and giants. ∆ log ε(Li, SGB−TOP) ≈ 0.12 shows nearly 2σ significance for the abundance up-turn. For titanium and calcium, shallow trends of weak significance ∆ log ε ∼ 1σ are seen. However, titanium additionally shows the aforementioned indicator of radiative levitation with a dip at the SGB, at significance 1σ as compared to 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 (out of the 384 available) processors and roughly 1.5 GB of memory. Following the evolution from the pre-main sequence stages up to the turnoff point (corresponding to ∼ 10 Gyr) takes roughly 12 hours. The final evolution into the RGB stage (corresponding to ∼ 1 Gyr) requires significantly more computation and possibly even manual intervention. The degree of stratification determines the required distribution and number of layers in the model, and the largest possible timestep able to result in convergence. The process of finding the correct settings is iterative: the wrong parameters will not result in convergence, requiring manual intervention to redefine them. As the abundance distribution changes, so do the required parameters! We repeat the comparison of Korn et al. (2007) between the observational interpretation and the stellar models with diffusion at an age of 13.5 Gyr, shown in Figure 3.5. Overall, the T6.0 models (log T0 = 6 in (2.1)) fit the observations best, matching each trend to within 1σ. The T6.09 model is somewhat flat, with deviations of ∼ 1σ for magnesium and iron. T5.8 typically deviates by 2σ from observed trends. The lithium up-turn at the SGB by nearly 2σ is reproduced only by the T6.0 model. T6.09 and T5.8 are much too flat and much too strong at ∼ 1.5σ in the respective direction. The case of the RGB outliers indicates a somewhat lower diffusion efficiency than do the other trends, excluding the T6.09 model, and putting T6.0 at ∼ 1.5σ. As suggested in Korn et al. (2007), a somewhat lower cluster age than 13.5 Gyr is expected. For a cluster age of 12.5 Gyr (Figure B.3), predicted behaviors are very similar to the 13.5 Gyr models – such a small change in the cluster age does not affect the abundance trends much, save for a slight flattening effect

35 Figure 3.5: Comparison of observations to predictions from diﬀusion models for a cluster age of 13.5 Gyr. Tx.x is shorthand for isochrones using turbulent mixing parameter log T0 = x.x. The T6.0 isochrone is found to match observations 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 to the analysis. As before, T6.09 shows a similar but flatter behavior to the higher ages, again showing weak trends for iron, magnesium, titanium and lithium, off by between 1σ and 1.5σ. T6.25 is much too flat for all but the calcium trend. Magnesium and iron are much too flat, and titanium does not show a sufficient downturn at the SGB. For lithium, the TOP–SGB behavior is reversed 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 partially destroyed – seen also as a general decrease in the lithium abundance by more than 0.1 dex.

3.3.4 Comparison to photometry The results would be strengthened by independent confirmation using other methods. 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 empirical calibrations covering the giant and dwarf stages. Casagrande et al. (2010) provides an updated empirical calibration covering only the TOP, and barely the SGB (the sample is quite thin at log g ∼ 3.6, for natural reasons). These empirical calibrations connect photometric observations to accurate ”true” temperatures. For each calibration, we use a metallicity of [Fe/H] = −2.0. Onehag¨ et al. (2009) provides theoretical photometric temperatures, from integration of synthetic spectra using model atmospheres similar to those used in the analysis – MARCS rather than MAFAGS-ODF. This instead can be

36 Figure 3.6: Comparison of observations to predictions from diffusion models for a cluster age of 11.5 Gyr. All models use the same initial elemental abundance – note the large depletion of lithium for the T6.25 isochrone. thought of as a way of bridging photometry and spectroscopy, by performing a spectroscopic estimate using photometric data. These temperatures should be considered consistent within the framework of spectroscopy – an extra way of checking that our deduced temperatures are indeed what the observations tell us rather than the product of (sub?)conscious modifications meant to support our initial suspicions. At this point, one must choose a philosophy: which temperature scale should 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 measurements of the distance coupled to estimates of the bolometric luminosity give the effective temperature by its very definition. On the other hand, the temperatures found by spectroscopic means are by definition consistent within a spectroscopic analysis. The synthetic temperatures 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 temperature requires that lines of a species at varying excitational energies all indicate the same abundance. The result for iron lines will typically be lower temperatures than those derived using Balmer lines. However, the excitation equilibrium is known not to be trustworthy due to 3D effects. The steeper temperature gradient of a 3D model atmosphere results in a cooler outer atmosphere, which affects the excitational populations in the Boltzmann equation. The temperature structure of a 1D model atmosphere results in too low populations in the low-energy excitational levels, resulting in an overestimate of the required abundance for these lines. Comparing the range of temperatures as deduced by photometry and spectroscopy, each calibration indicates (by flat or slight upward slopes) that the spectroscopic temperature scale is correct, or that it shows too wide a range of

37 Figure 3.7: Comparison of spectroscopic effective temperatures and all photometric indices for each calibration, as listed in Table B.3. IRFM 2010 refers to 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). A flat curve indicates only an overall absolute shift in temperatures. A negative slope indicates a wider span of effective temperatures, while a positive slope indicates a narrower span. temperatures. Compressing the temperature scale would typically strengthen the observed abundance trends. Our temperature scale is therefore a conservative estimate. What about the absolute temperatures at the upper end of the temperature scale? The old set of calibrations indicates lower TOP temperatures than deduced from spectroscopy. The new IRFM calibration, Casagrande et al. (2010), however suggests roughly 100 K higher TOP temperatures, with quite good agreement between the three color indices. Two of three indices agree on even larger increases for the SGB temperatures, but these lie at the very edge of validity for the calibration (see their figure 13). A larger temperature increase at the SGB than TOP would strengthen e.g. the lithium trend, but again we keep to a conservative estimate. The fact that this calibration typically indicates somewhat higher temperatures than spectroscopic values for metal-poor stars is noted by its authors. Since the calibration does not yet reach giant stars, the implication for the entire temperature scale is unknown. There are however indications that the previous results for giants are correct, as these already were derived from a large sample of suitable stars.

38 Chapter 4

Results

4.1 Examining diﬀerent temperature scales

If, as a qualified guess, we assumed that the RGB temperatures of the original analysis were indeed correct, but the TOP temperatures were underestimated by 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 expanding it. The simple method is a linear expansion of the entire scale, giving roughly 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) differs by only 20 K (Korn et al., 2007), but a peak in the temperature correction is seen near the SGB. Hence, the broadening theory of Barklem et al. (2000) would expand ∆Teff(TOP − SGB), thus raising the inferred abundances at the TOP. This would for example flatten the abundance trend of lithium somewhat. The precise correction for the temperature of the SGB stars using this broadening theory is yet unknown. The sample of Lind et al. (2008) includes stars photometrically similar to those of our SGB sample. Stars 6760, 13466, 16588, 506139, and 502729 in their results show a mean temperature 5834125 K, i.e. a mere 30 K hotter than our results. Such a small correction is insignificant for the lithium trend. Thus, we stick to the linear method of expanding the temperature 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 an implicit intention of keeping the other temperatures fixed. Korn et al. (2007) argues against such a shift on the grounds of its severity: could the error in temperature between TOP and SGB stars really be 30 %? They further note that while such a shift would indeed lessen the trends observed for most elements, it would not be sufficient to remove the magnesium trend. If one were to deviate from the spectroscopically consistent temperatures found, how would one handle the surface gravities? Spectroscopically, the ionization equilibrium condition is applied: the neutral and ionized species of iron should indicate a consistent abundance. Since we already deviate from the spectroscopically indicated temperatures, we add an additional gravity-scale where surface gravities are instead read off from a theoretical isochrone. The distinction affects primarily majority species – in our analysis, this means singly ionized 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 fetch the isochrone log g values for an age of 12.5 Gyr. In Teff − log g space, the T6.0 and 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 main mechanism of atomic diffusion altering the rate of evolution. See Figure B.1 for a comparison of the isochrones at different ages – log g depends only slightly on age. As the temperature scale already deviates from the spectroscopic findings, we do not expect the ionization equilibrium condition to fare better. Thus, we choose the set of isochrone log g values for our primary analysis, and call this scale +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? Following Bonifacio et al. (2007), increasing the difference between TOP and SGB stars by roughly 100 K would nearly remove the trends for iron, calcium, and titanium, as well as somewhat flattening the TOP–SGB trend for lithium. To overall flatten the trends, the difference between TOP and RGB stars would need to be increased by roughly 200 K. Encompassing both these requirements, we construct yet another temperature scale as a linear expansion by 200 K rather than 100 K, resulting in an increase of ∆Teff(TOP − SGB) by roughly 80 K. Revisiting the photometric temperatures in Figure 3.7, we find no agreement with such a temperature scale. It should therefore be considered mainly an experiment in what one might achieve by investigating the very boundaries of the confidence interval. Curiously, this temperature scale resembles that of Gratton et al. (2001), with TOP stars roughly 220 K hotter, but bRGB stars at roughly the same temperature. It is however convincingly shown in Korn et al. (2007) that the high TOP temperatures are probable artefacts of échelle-blaze residuals. Figure 4.1 shows 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 to reproduce the wings of Hα, with a deviation somewhat larger than 2σ deduced from the noise level. This is however dependent on the broadening theory employed (we employ Ali and Griem, 1966). Barklem et al. (2000) generates overall stronger lines, which in a differential analysis somewhat cancel, thus giving a similar result. Yet again, we choose two sets of surface gravities: one in a consistent spectroscopic manner, and one from comparison to theoretical isochrones in Fig- ure B.1. With the 200 K expansion of the +200 temperature scale, our TOP stars are found at a reasonable evolutionary stage only for the youngest isochrone 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, in conflict with cluster photometry (Figure 3.1). Again, we detail +200 in the primary analysis, and show +200e only in Appendix A. Our new temperature scales 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 g connections consistent with the 13.5 Gyr traditional stellar model, as shown in the right hand panel of Figure B.1. To find the corrections to the original results, a reduced analysis is performed. For iron, four representative weak lines each are chosen for the neutral species, and five for the singly ionized species. For neutral iron, they are chosen at varying excitation energies, ranging from 0.1 to 4.3 eV, to encompass the range of lines’ indicated abundances when the excitation equilibrium condition is not fulfilled. For the other elements, the full (albeit rather short) line list is used. The full list of lines used in the analysis at the new temperature scales is 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 the lowest S/N in the TOP and SGB groups respectively. Lines are synthesized at the original temperature scale (orig; top, red lines), as well as the new scales +100 (middle, green lines) and +200 (bottom, blue lines). While +100 still reproduces the observed spectral line, +200 is obviously too strong for both stars.

41 Table 4.1: The new temperature scales, each with two sets of surface gravities. 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 0.11 SGB 5865 3.73 60 0.15 bRGB 5486 3.46 30 0.09 RGB 5130 2.56 0 0.0 +100e – Spectroscopic (ionization equilibrium) log g TOP 6354 4.09 100 0.20 SGB 5865 3.71 60 0.13 bRGB 5486 3.46 30 0.09 RGB 5130 2.56 0 0.0 +200 – 11.5 Gyr isochrone log g TOP 6454 4.04 200 0.15 SGB 5925 3.73 120 0.15 bRGB 5526 3.47 70 0.10 RGB 5130 2.56 0 0.0 +200e – Spectroscopic (ionization equilibrium) log g TOP 6454 4.24 200 0.35 SGB 5925 3.80 120 0.22 bRGB 5526 3.47 70 0.10 RGB 5130 2.56 0 0.0 dance is given only by the ionized species, Fe ii. For the temperature scale using ionization equilibrium log g, both ionization stages per definition indicate the same abundance.

4.2 Examining an expanded temperature scale

Here 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 results with 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 trend has shrunk to ∆ log ε(Mg) = 0.19 dex, significant at 2.7σ – or 3.9σ using the least polluted RGB outliers. The overall trend for calcium has flattened completely. 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 flattened to insignificant levels, off by just 1.1σ at the SGB down-turn. Thus, at this temperature scale, the magnesium and iron trends still holds at large significance. Calcium, titanium and lithium flatten to roughly within agreement with traditional stellar models.

42 Figure 4.2: Comparison of abundance trends at the temperature scale +100 to 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 agreement with observed trends. Magnesium, iron and lithium are perfectly reproduced. However, calcium and titanium trends show weak radiative levitation, and deviate by up to 1.5σ. T6.09 fares slightly worse for calcium and titanium at up to 1.7σ. T5.8 matches the titanium trend nicely, but shows trends too strong in all other cases, disagreeing up to 3σ for iron. The lithium trend however is reproduced at a strongly upward shifted initial abundance. This seems a spurious result of the strong slope of the SGB-upturn, which is somewhat offset in temperature. A slight shift in temperature will dramatically change the inferred initial abundance and level of agreement. The magnesium abundance of the 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 T6.0 reproduces the trend of each element even better, though still somewhat weak for titanium at 1.2σ. T6.09 fares somewhat worse for calcium and titanium, off by up to 1.5σ. The magnesium abundance of the RGB outliers shows excellent agreement with T6.0. At an age of 11.5 Gyr (Figure B.5), T6.09 overall seems better than for the older isochrones, within the error bars for magnesium, calcium, iron and lithium, but slightly off by 1.3σ for titanium. T6.25 reproduces the calcium and iron trends, but fails similarly for magnesium and titanium at ∼ 1.4σ. The lithium trend, TOP–SGB, is inverted, off by 2.5σ. The magnesium abundance of 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 scale labeled +200 in Table 4.1. See Appendix A for the results with log g values deduced from the ionization equilibrium of iron (+200e).

43 4.3.1 Abundance trends As expected, the expanded temperature scale causes the trends to overall flatten. Magnesium however retains a signature at the 2σ level – or 3σ for the RGB outliers. Lithium (TOP−SGB) and calcium both flatten to nearly insignificant trends. Titanium and iron show barely significant ∼ 1.5σ signatures for the TOP–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 isochrones do not reach the alleged TOP temperatures of our stars. As we do wish to compare the behavior anyway, we assume a slight shift in the isochrone temperatures (applied hypothetically – for the models, this corresponds to slightly altered model boundary conditions). The T5.8 models show the correct 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 iron shows nearly 4σ deviation. T6.0 matches magnesium and lithium, but fails for calcium and titanium at more than 2σ, and iron at 1.4σ. T6.09 matches magnesium and lithium nicely, as well as iron, but fails for calcium and titanium at > 2σ. The magnesium abundance of the RGB outliers is matched only by the T6.0 model. At 12.5 Gyr (Figure B.7), T6.0 and T6.09 match iron, lithium and magnesium. The RGB outliers are matched by both sets of models. Calcium and titanium are similarly oﬀ at ∼ 2σ. At 11.5 Gyr, results are shown in Figure 4.4. T6.25 and T6.09 match iron and magnesium. The RGB outliers match only T6.09. Calcium and titanium fail similarly at ∼ 2σ for both models. Lithium matches T6.09, but fails at ∼ 2σ for T6.25.

4.4 Chromium and nickel

An effort was made to include additional elements in the analysis. For this, NIST was queried for several elements with atomic diffusion abundance trend predictions available, for any ionization stage. Since metal lines in general are weaker in dwarfs (due to the higher temperature and stronger surface gravity), the spectra of TOP stars decide which lines are possible to detect. For simplicity of identification, the metal-poor field star HD 84937 was used as a proxy for the 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 lines of the ionized species would have been preferable due to their small NLTE and 3D effects (Asplund, 2005). For chromium, all three Cr i lines of multiplet 7 were detected. Under the philosophy of quality over quantity, we discard all but λ5206 due to blends with Fe 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 in future 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 +200 to predictions from diﬀusion models for a cluster age of 13.5 Gyr.

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

45 the solar abundance. Therefore, one should expect a displacement in absolute abundances. The results of our LTE analysis are presented in full in Table B.4, as well as along with diﬀusion model predictions in Figure B.2.

4.4.1 NLTE and 3D Chromium A complex interplay of NLTE mechanisms is noted by Berge- mann (2010); Bergemann and Cescutti (2010). Most importantly at low metallicity, superthermal UV radiation along with insufficient collisional interactions result in overionization and photon pumping. The former mechanism connects to temperature, while the latter connects to metallicity and surface gravity, so corrections are found to be larger for giants than dwarfs. NLTE corrections are on the order of +0.3 dex for metal-poor dwarfs, but no values are given for giants. 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 order of the discrepancy between our observations and predictions – which notably show slopes of reverse signs. Since NLTE effects are sometimes enhanced and sometimes counteracted by 3D effects, it is yet unknown what to expect from 3D-NLTE considerations. We perform our analysis in LTE, in the hope that corrections from 3D or NLTE analyses may later be applied, allowing a straightforward comparison to our results.

Nickel As another iron-peak element, we expect similar NLTE effects for neutral nickel as for the neutral species of iron and chromium. However, no detailed calculations seem available for nickel. Consequently, we perform our analysis in LTE, in the hope that corrections may become available in the future. If one were to use our results as a benchmark, Figure B.2 shows excellent agreement for the SGB–RGB trend with all available isochrones. The TOP abundance implies either weak turbulent mixing or stronger NLTE effects than the other evolutionary stages do. Specifically, the decreased abundance corresponds qualitatively to overionization, in line with expectations for iron-like elements.

46 Chapter 5

Conclusions

A lower cluster age results in somewhat flatter trends in the theoretical atomic diffusion predictions. Flattened trends may in turn be compensated by a somewhat lower turbulence efficiency, which allows a greater net effect of gravitational settling. However, this relation is inverted for calcium and titanium, where radiative levitation raises their abundances at the TOP. There, weaker turbulence allows radiative levitation to flatten the TOP–RGB behavior. Con- sequently, each element shows a specific dependence on the strength of turbulence in both sign and amplitude. In other words, the diffusion signature is not perfectly degenerate as regards age and strength of turbulence.

5.1 The original temperature scale

In our comparisons with observations interpreted at the original temperature scale, we find that amongst our sample of theoretical isochrones, an age of 13.5 Gyr with a turbulence strength of T6.0 gives the best reproduction of the observed abundance trends. This is in line with the conclusions of Korn et al. (2007). A slightly lower efficiency of turbulence should strengthen and thus reproduce the trends even better. Our larger sample of theoretical models allows us to deduce that a lower age, 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 weaker turbulence is exacerbated by the raised magnesium abundance of the RGB outliers. From the general behavior of the isochrones, by means of linear interpolation, we find roughly the 1σ limits for a cluster age of 13.5 Gyr at log T0 ∈ [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 titanium do not contribute to the upper limit. For magnesium, the RGB outliers exacerbate the need for a low log T0 value, with a 1σ upper limit at log T0 ≈ 5.95, compressing the allowed range for an overall ∼ 1σ correspondence to roughly log T0 ∈ [6.0, 5.9]. At a cluster age of 11.5 Gyr, the range is shifted somewhat downward, with an estimate of log T0 ∈ [6.0, 5.85], with the upper and lower limits set most stringently by the iron and calcium trends. For magnesium, the RGB outliers set an upper limit of log T0 ≈ 5.95, giving roughly the range log T0 ∈ [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 estimate of the overall degenerate behavior of strength of turbulence and cluster age is shown in Figure 5.1.

47 Figure 5.1: The degenerate behavior of age and strength of turbulent mixing, from tentative linear extrapolation at the original temperature scale, orig. Filled squares denote the available isochrones. Horizontal bars show roughly the upper and lower 1σ limits for the overall behavior of the trend. The shaded area shows the parameter space when considering the magnesium trend of the RGB outliers – an effort to lessen the impact of pollution. The vertical dashed line 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 model preferred for field stars (Meléndez et al., 2010).

Elemental abundance trends at this temperature scale are in tentative agreement with the cluster age 11.5 Gyr. Considering Figure B.1, the TOP stars are some 300 K cooler for the original temperature scale than the turnoff point. This is in disagreement with cluster photometry, which indicates a difference of 100–200 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 Gyr deduced using the white dwarf cooling sequence, Hansen et al. (2007) – seems the best option. Additional isochrones with low efficiency of turbulence at this age are required to confirm or dismiss this conclusion.

5.2 The extended temperature scale

From a diffusion perspective, calcium and titanium still demonstrate slight radiative levitation signatures. We find the best correspondence for the 12.5 Gyr T6.0 model, Figure 4.2. A somewhat lower turbulence efficiency than log T0 = 6 should give the best correspondence 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 log T0 value. At 13.5 Gyr, with T6.0 still within the error bars of the iron trend, it should be possible to find a value log T0 ∈ [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 the temperature of the turnoﬀ point in the 13.5 Gyr isochrones (in strong conﬂict with cluster photometry). As the abundance trends show rapid variation with temperature at this evolutionary stage, a quite large uncertainty is introduced in our extrapolation of the behavior at lower values of log T0. Thus, the uncertainty in these estimates should be considered as greater than for the previous temperature scale.

At 12.5 Gyr, we ﬁnd the 1σ limits log T0 ∈ [6, 5.9], with quite large uncertainty in the lower value. The strictest upper limit is set by the radiative levitation signature of titanium (a dip at the SGB), and the lower is hinted at by the ﬂat trends of calcium and iron. The magnesium trend of the RGB outliers does not restrict this range further.

At 11.5 Gyr, linear interpolation sets limits of roughly log T0 ∈ [6.1, 5.85], with considerable uncertainty in both limits. The upper limit is set most strictly by titanium (at log T0 ≈ 6), and the lower by magnesium. Considering the trend of the RGB outliers, the limits are shifted to log T0 ∈ [6, 5.8]. The resulting relations are shown in Figure 5.2.

For a cluster age of 11.5 Gyr, the TOP stars are some 200 K cooler than the turnoff point (see Figure B.1), in agreement with the cluster photometry. Additionally, elemental abundance trends are found in tentative agreement with the atomic diffusion models of low efficiency of turbulence, when shifted toward this 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 are required to confirm or dismiss this conclusion.

49 Figure 5.3: The deduced initial lithium abundance (dashed line), as found at 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 signature

At the +200 temperature scale, trends become inconsistent, as shown in Fig- ure 4.3. Calcium and titanium show signs of radiative levitation, while iron and magnesium flatten. The former indicate weak turbulence, while the latter require a stronger setting. Additionally, the nearly flattened lithium trend, still hinting at an SGB upturn, indicates a strength somewhere between the titanium and magnesium requirements. Thus, no single diffusion model is able to reproduce all trends – different elements indicate different strengths of turbulence, so the upper and lower limits to log T0 do not overlap. The remaining strong trend for the magnesium abundance at 2σ, as well as the titanium and iron trends at 1.5σ also rule out traditional stellar models at similar significance as for the diffusion models. This should come as no surprise, as we have already ruled out this temperature scale on the basis of its disagreement with Hα fitting and photometry. Rather than a breakdown of diffusion model predictions, one could regard it as proof that the free parameter description of turbulence indeed does not allow the prediction of just any arbitrary trends.

5.4 Implications for the primordial lithium abundance

Figure 5.3 shows the deduced primordial lithium abundance of the best model for the temperature scales orig and +100. With an uncertainty of 0.06 dex in the predicted primordial abundance, and 0.10 dex in the detected absolute lithium abundance (as adopted from Korn et al., 2007), the comparison has an uncertainty σ ≈ 0.12 dex. This puts the estimated initial lithium abundance at the original temperature scale (orig) log ε(Li) = 2.48 0.1 in disagreement by an amount of 70 %, a discrepancy at the 2σ level with theoretical expectations of log ε(Li) = 2.71 0.06 (Cyburt et al., 2010). For the expanded temperature scale (+100), the estimated initial lithium abundance log ε(Li) = 2.55 0.1 is found in milder disagreement by an amount of 45 %, a discrepancy of roughly 1.5σ.

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 original temperature scale, and falls just within the 1σ error bars for the expanded temperature scale. Altering the age of the cluster seems insignificant for the initial lithium abundance – for our sample of T6.09 isochrones, a shift of < 0.02 dex Gyr−1 is found, and the T6.0 isochrones indicate the same initial abundance for ages of 12.5 and 13.5 Gyr. However, the efficiency of turbulence is found to strongly determine the TOP–SGB behavior. Thus, the correspondence between age and optimal efficiency of turbulence together give a range of possible initial abundances compatible with the spectra investigated. Additionally, the assumed temperature scale plays a very large role due to the overall shifted lithium abundances as the temperatures of TOP and SGB stars increase. It should be noted that +200, the very most extreme scale specifically constructed to downplay the importance of atomic diffusion, results in the flattest trends. These are found to be reasonably (although not nearly sufficiently) consistent with the strong turbulence model T6.25, which results in the very largest 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 turns out to be its allied. Following the behavior of Figures 5.1 and 5.2, a decrease in the assumed cluster age couples to a decrease in the required efficiency of turbulence – quite roughly for +100. As models of low efficiency of turbulence show a stronger lithium abundance trend comparing TOP–SGB, they correspond to a somewhat raised initial abundance. Thus, for a cluster age of 11.5 Gyr, we expect somewhat higher initial lithium abundances than deduced for the available set of isochrones. Hence, the deduced ∼ 1.5σ disagreement between the Big Bang Nucleosynthesis value from WMAP data, and our observed value at the +100 temperature scale should be expected to decrease slightly if suitable theoretical isochrones are used in the analysis. Additionally, as the effects of 3D and NLTE somewhat cancel for lithium, one should expect a slight upward correction from our NLTE abundances. Considering the Spite plateau, the fact that the lithium surface abundance depletion shows only weak correspondence to age is important. Otherwise, one could not reproduce the required flat and thin behavior for a stellar sample with some variation in age. Additionally, there is some constraint in the efficiency of turbulence, which must result in a plateau which is flat for stars near the turnoff point. For instance, Korn et al. (2007) note that the weak mixing of isochrone T5.8 at 13.5 Gyr does not fulfill this constraint. Our results should be compatible with this constraint, as they tentatively suggest a compatible range of values somewhere roughly between the models T5.8 and T6.0 at a cluster age of 11.5 Gyr for the +100 temperature scale.

Chapter 6

Summary

From exploring two expanded temperature scales, at two sets of log g each, we conclude that it is not possible for all observed abundance trends to be ex- plained simply as artefacts of the temperature scale. Flattening the titanium trend would require a decrease in ∆Teff(TOP − SGB), which at the same time would only strengthen the magnesium and lithium trends. Flattening the magnesium trend would require expanding ∆Teff(TOP − RGB) by roughly 400 K, in conflict with all temperature indicators, as well as cause the other elemental trends to steepen yet again. Hence, we expect that the expanded temperature scales +100 and +200 show the overall flattest abundance trends. +100 matches the cluster photometry and expected abundance trend predictions for stellar models with atomic diffusion at an age of 11.5 Gyr, in line with the external prior of Hansen et al. (2007). An additional set of theoretical isochrones for this age is required to finalize the analysis. Regardless of temperature scale, predicted abundance trends match observations only with low efficiency of turbulence. The setting log T0 = 6.25, found by Meléndez et al. (2010) to be in agreement with their sample of metal-poor field stars, does not reproduce our observations. Notably, such high efficiency of turbulence causes additional depletion of lithium, which inverts the TOP–SGB trend, at large significance. The initial lithium abundance corresponding to model predictions describing our observed abundance trends is expected to be similar to the data in Figure 5.3, discrepant from BBN-WMAP expectations by 2σ for orig, or 1.4σ for +100. The final selection of the optimal theoretical isochrone of +100 might shift 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 +100 in at least 1σ agreement with BBN-WMAP predictions. Errors in modelling assumptions could mask shortcomings in the predictive power of atomic diffusion models. As 3D-NLTE models become available, no doubt will the results of our NLTE analyses change, albeit slightly. It would however seem contrived to blame the observed abundance trends, with their remarkably good correspondence to predictions, on modelling shortcomings. Korn et al. (2007) conclude that as 3D effects are expected to affect elements quite differently by both sign and magnitude, the abundance trends are not artefacts of 1D modelling.

6.1 Ongoing and suggested future eﬀorts

We have derived eﬀective temperatures using the broadening theory of Ali and Griem (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 for the 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 effect on this temperature difference and its resulting abundance trends should be investigated in future analyses of the present stellar sample. Lind et al. (2008, 2009b) are textbook examples in (at least) two respects of how a large-scale study of atomic diffusion should be performed. They use GIRAFFE for investigations covering the evolution from TOP to RGB, for as many elements as possible. Lind et al. (2009b) observed > 300 stars, for a tally of 454 including archive data, sufficient for statistical identification of unpolluted cluster members. Lind et al. (2008) observe a smaller sample of > 100 stars, but use four wavelength ranges covering suitable lines of four elements. Lind et al. (2009b) as well as González Hernández et al. (2009) include main sequence 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 a low efficiency of turbulence. The 3D-NLTE results of Gonzalez´ Hernandez´ et al. (2009) however suggest that the main sequence behavior of the theoretical models is erroneous, as quantitative agreement is found only for models of strong turbulent mixing – specifically, T6.25. This discrepancy should be further investigated. The metallicity dependence is not yet understood. The evolutionary dependence 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 Meléndez 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 field stars at [Fe/H] . −3 (Sbordone et al., 2010). By narrowing down the range of circumstances where different strengths of turbulence reproduce the observations, perhaps the physical nature of turbulent mixing 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 are detected in our spectra, but lack theoretical predictions – although scandium is assumed to behave similarly to calcium and titanium. The barium abundances detailed in Korn et al. (2007) show no clear trend, with equal values at the TOP and RGB. Such non-trends are of course also a result, and require confirmation in theoretical predictions.

54 Appendix A

Abundance trends using the ionization equilibrium

A.1 Examining an expanded temperature scale

For +100e compared to +100, the iron trend flattens to 1.7σ while the titanium trend reaches a slightly larger SGB down-turn ∼ 1.5σ. As the titanium trend strengthens while the iron trend flattens, they each restrict the upper and lower limit respectively to the turbulence efficiency. Among the 13.5 Gyr models, shown in Figure A.1, T6.09 reproduces the iron trend, while T5.8 reproduces the titanium trend. T6.0 seems the best compromise 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 reproduce the 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 the iron trend well. Titanium is off by 1.7σ for both.

A.2 Attempting to remove the diﬀusion signature

For the temperature scale +200e, the iron trend flattens completely while the titanium trend is enhanced to the 2σ level. Thus, two trends – magnesium and titanium – remain irreconcilable with traditional stellar models at the 2σ level. Expanding the temperature scale further will exacerbate the problem for titanium by means of unphysical log g values, and start shifting the currently flattened abundance trends toward negative slopes. No model (Figures A.3, B.8, and B.9) reproduces the titanium trend better than ∼ 3σ. For iron, the flat trend matches only the 11.5 Gyr isochrones, with the best match for T6.25.

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

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

Figure A.3: Comparison of abundance trends at the temperature scale +200e to predictions from diﬀusion models for a cluster age of 13.5 Gyr. Other elemental 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 − VV − I TOP ...... 9655 16.200 0.311 0.688 0.396 0.578 10197 16.160 0.314 0.695 0.402 0.585 12318 16.182 0.312 0.691 0.399 0.581 506120 16.271 0.308 0.683 0.391 0.570 507433 16.278 0.308 0.682 0.391 0.570 SGB...... 5281 15.839 0.368 0.796 0.476 0.686 8298 15.832 0.369 0.799 0.478 0.689 bRGB ...... 3330 15.227 0.428 0.932 0.568 0.784 6391 15.551 0.412 0.890 0.539 0.760 15105 15.439 0.419 0.909 0.551 0.770 23267 15.339 0.424 0.920 0.560 0.778 500949 15.514 0.416 0.900 0.544 0.764 RGB ...... 4859 13.815 0.470 1.028 0.636 0.857 7189 13.729 0.474 1.038 0.641 0.862 11093 13.551 0.478 1.052 0.651 0.875 13092 13.644 0.478 1.047 0.646 0.868 14592 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 new temperature scales.

Species Line(s) Treatment Fe i λλ5217.3, 5225.5, 5424.0, 6421.3 NLTE Fe ii λλ4923.9, 5018.4, 5169.0, 5316.6, 5234.6 LTE Ti ii λλ5188, 5226 LTE Ca i λλ6222, 6162, 6439 NLTE Li i λ6707 NLTE Mg i λ5528 NLTE Ni i λ5476 LTE Cr i λ5206 LTE

57 Figure B.1: Comparison of theoretical isochrones at diﬀerent ages. Data points represent the original spectroscopic temperature scale orig, the +100 and +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 temperature scale orig compared to the full set of isochrones. Corrections for the alternative temperature scales are given in Table B.5

58 Table B.3: Photometric eﬀective temperatures, also shown in Figure 3.7

Group Star ID Spectroscopic Hα IRFM a IRFM b MARCSc b − y v − y B − VV − I b − y B − VV − I b − y v − y TOP ...... 9655 6260 6227 6212 6159 6128 6371 6372 6320 6281 6265 10197 6250 6204 6187 6135 6096 6349 6343 6283 6232 6232 12318 6240 6217 6201 6148 6113 6363 6358 6304 6265 6248 506120 6260 6248 6233 6179 6163 6393 6396 6363 6331 6331 507433 6260 6249 6235 6180 6166 6393 6396 6363 6331 6331 SGB...... 5281 5800 5791 5819 5822 5682 5954 6008 5810 5839 5853 8298 5810 5802 5828 5832 5693 5947 6000 5798 5839 5853 bRGB ...... 3330 5430 5351 5359 5491 5281 ...... 5507 5500 6391 5510 5437 5452 5591 5351 ...... 5569 5562 15105 5470 5399 5411 5551 5321 ...... 5543 5540 23267 5370 5373 5385 5519 5298 ...... 5525 5522 59 500949 5500 5418 5431 5574 5338 ...... 5558 5551 RGB ...... 4859 5150 5139 5153 5272 5081 ...... 5245 5245 7189 5130 5120 5134 5257 5066 ...... 5229 5229 11093 5100 5102 5104 5224 5034 ...... 5194 5194 13092 5120 5103 5115 5242 5051 ...... 5207 5212 14592 5130 5113 5126 5251 5060 ...... 5220 5224 502074 5150 5148 5162 5279 5088 ...... 5254 5254

TOPavg ...... 6254 6229 6213 6160 6133 6374 6373 6327 6288 6281 SGBavg ...... 5805 5797 5824 5827 5688 5951 6004 5804 5839 5853 bRGBavg ...... 5456 5396 5408 5545 5317 ...... 5541 5535 RGBavg ...... 5130 5121 5132 5254 5063 ...... 5225 5226 ∆(TOP − RGB) . 1124 1108 1082 906 1070 ...... 1063 1055 ∆(TOP − SGB) . 449 433 390 333 446 423 369 523 449 428 a Empirical calibration (Alonso et al., 1996, 1999) b Empirical calibration (Casagrande et al., 2010) c Theoretical ﬁt (Onehag¨ et al., 2009, Onehag¨ 2010, private communication) Table B.4: Spectroscopic parameters and abundances for individual stars as well as group averages. Group averages are averages of the eﬀective temperature, surface gravity, microturbulence and metallicity, while the elemental abundances are derived by applying the group average stellar parameters to averaged line spectra. The original temperature scale is used, details given in Section 3.3.

Group Star ID Teﬀ log g ξ [Fe/H] log ε(Li) log ε(Mg) log ε(Ca) log ε(Ti) log ε(Ni) log ε(Cr) log ε(Na) (K) (cgs) (km s−1) NLTE LTE NLTE NLTE LTE LTE LTE LTE TOP ...... 9655 6260 3.85 2.1 -2.29 2.24 5.63 ...... 10197 6250 3.95 2.0 -2.29 2.24 5.58 ...... 12318 6240 3.90 2.0 -2.28 2.15 5.64 ...... 506120 6260 3.85 1.9 -2.26 2.32 5.67 ...... 507433 6260 3.90 2.0 -2.26 2.32 5.67 ...... SGB...... 5281 5800 3.55 1.75 -2.25 2.40 5.75 ...... 8298 5810 3.60 1.75 -2.23 2.32 5.75 ...... bRGB . . . . . 3330 5430 3.35 1.7 -2.16 1.27 5.66 ...... 4.39

6391 5510 3.40 1.6 -2.14 1.54 5.93 ...... 4.39 60 15105 5470 3.40 1.75 -2.18 1.40 5.88 ...... 4.45 23267 5370 3.30 1.8 -2.23 1.25 5.83 ...... 4.35 500949 5500 3.40 1.8 -2.19 1.45 5.75 ...... 4.49 RGB ...... 4859 5150 2.65 1.6 -2.12 0.90 5.89 ...... 4.16 7189 5130 2.55 1.6 -2.14 0.96 5.81 ...... 4.36 11093 5100 2.50 1.7 -2.14 1.09 5.96 ...... 3.94 13092 5120 2.55 1.6 -2.12 0.89 5.88 ...... 4.25 14592 5130 2.55 1.6 -2.12 1.08 5.96 ...... 4.06 502074 5150 2.60 1.5 -2.10 1.09 5.75 ...... 4.24

TOPavg .... 6254 3.89 2.0 -2.28 2.24 5.65 4.53 2.99 3.72 3.33 ... SGBavg .... 5805 3.58 1.75 -2.24 2.36 5.75 4.52 2.91 3.88 3.26 ... bRGBavg .. 5456 3.37 1.73 -2.18 1.38 5.79 4.55 3.01 3.91 3.22 ... RGBavg .... 5130 2.56 1.6 -2.12 0.98 5.88 4.60 3.05 3.94 3.27 ... Table B.5: Corrections to the group averages of Table B.4 for the alternative temperature scales introduced in Sec- tion 4.1, and analyzed in Section 4.2 and Section 4.3. The employed line list is given in Table B.2.

Group ∆Teﬀ ∆ log g ∆ log ε(Fe) ∆ log ε(Li) ∆ log ε(Mg) ∆ log ε(Ca) ∆ log ε(Ti) ∆ log ε(Ni) ∆ log ε(Cr) +100 – 12.5 Gyr isochrone log g TOP 100 0.11 0.05 0.07 0.04 0.055 0.075 0.08 0.085 SGB 60 0.15 0.06 0.045 0.025 0.025 0.075 0.055 0.053 bRGB 30 0.09 0.025 0.025 0.015 0.01 0.035 0.03 0.03 RGB 0 0 0 0 0 0 0 0 0 +100e – Spectroscopic log g TOP 100 0.20 0.075 0.07 0.04 0.055 0.10 0.08 0.085 SGB 60 0.13 0.05 0.045 0.025 0.025 0.07 0.055 0.053 61 bRGB 30 0.09 0.025 0.025 0.015 0.01 0.035 0.03 0.03 RGB 0 0 0 0 0 0 0 0 0 +200 – 11.5 Gyr isochrone log g TOP 200 0.15 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 bRGB 70 0.10 0.047 0.06 0.04 0.045 0.065 0.075 0.08 RGB 0 0 0 0 0 0 0 0 0 +200e – Spectroscopic log g TOP 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 bRGB 70 0.16 0.06 0.06 0.04 0.045 0.085 0.075 0.08 RGB 0 0 0 0 0 0 0 0 0 Figure B.3: Comparison of abundance trends at the original temperature scale 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 +100 to predictions from diﬀusion models for a cluster age of 13.5 Gyr.

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

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

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

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

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

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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 is completely homogeneous. Convection is accounted for by mixing length theory, and the free parameters micro- and macroturbulence.

3D Advanced stellar atmospheres calculated with hydrodynamical eﬀects describing convection. Leads to granulation, which changes line formation in a way models are able to reproduce with astonishing accuracy.

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

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

Boltzmann equation Describes the atomic populations at different excitational levels within one ionization stage, under the assumption of (local) thermodynamic equilibrium. bound-bound transition Excitation from one energy level to another. The energy difference between the levels is fixed, and so the transition occurs at a specific wavelength. bound-free transition Ionization. The electron is given enough energy to leave its atom altogether. Additional energy above the ionization limit is retained as kinetic energy, so the transition does not have an upper energy limit (corresponding to not having a lower wavelength limit). Radiation at any wavelength lower than the ”edge” of a bound-free transition may trigger 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 a larger volume at lower temperature, the small volume element experiences buoyancy. It will tend to rise until this is no longer the case. This causes granulation. departure coefficient The ratio of the level populations calculated under the NLTE LTE assumptions of NLTE and LTE, bi = ni /ni . dex A relative change given on the scale of logarithms to base 10. See Table B.6 for a few examples.

69 differential abundance [X/H] = log ε(X)? − log ε(X) , the amount of element 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 (subscript u) and lower (subscript l) level: Spontaneous deexcitation (Aul), radiative excitation (Blu) and deexcitation (Bul), and collisional excitation (Clu) and deexcitation (Cul). globular cluster A gravitationally bound system of stars of nearly identical ages (they formed at the same time), which by definition corresponds to the concept of an isochrone. A comparison between observations of a globular cluster and an isochrone describing that cluster should give a simple 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 radiate away its surplus of energy. As it grows colder, buoyancy is lost, and it sinks along the edges of the next rising element. This is seen as granulation: hotter (brighter) patches surrounded by colder (darker) filaments in a honey-comb or ”foamy” pattern. When observing an entire star at once, thousands of such granules are averaged into the same spectrum. Since hotter and colder elements result in different line shapes, the averaged line is not reproduced by 1D modeling from first principles.

Hα The most prominent of the hydrogen lines in the visible spectrum, at 6562.8 Å. The ﬁrst (α) in the Balmer-series of lines. isochrone A set of results from several stellar evolution models with varying mass parameters but otherwise identical in e.g. metallicity and (by deﬁnition) age.

Line formation Transitions between energy levels of an atom may emit (spontaneously or stimulated by another photon) or absorb a photon at a wavelength corresponding to the transition energy gap. The absorption mechanism drains photons from the speciﬁc wavelengths of the relevant transitions, forming an absorption line. By matching a synthetic spectrum to what is observed, the stellar parameters are deduced spectroscopically, within the framework of a model atmosphere and some appropriate line formation mechanisms.

LTE Local thermodynamical equilibrium. An adaptation of thermodynamic equilibrium to stellar atmospheres, where all processes are said to be local. The radiation intensity is set by the Planck function, and the atomic populations are set by the Saha and Boltzmann equations. The general 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 atomic number Z > 2, or a mass number A > 5.

70 metallicity Typically given as [Fe/H], the amount of iron as compared to the sun in logarithmic units. Deﬁned this way since iron lines dominate the typical visual spectrum, and iron with some exceptions seems to represent the overall amount of metals well. microturbulence A fudge-factor required in 1D atmospheres to synthesize saturated lines. Turbulence here refers only to small-scale motion, i.e. a local Maxwell-Boltzmann velocity distribution. model atmosphere A set of useful parameters given at the range of depths involved in line formation: e.g. temperature T , gas pressure Pg, electron pressure Pe, density ρ.

NLTE Calculations of atomic level behaviour without the assumption of local thermodynamical equilibrium (LTE) – hence non-LTE, NLTE. Atomic populations depend on eﬀects of the (non-local) radiation ﬁeld, which in turn 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 exponential removal of light along the line of sight. Larger numbers refer to greater depths, smaller numbers to the outer regions of the photosphere. Optical depth relates to the effective temperature as Teff = T (τ = 2/3) in the so-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 or excitational level, as compared to the total amount of atoms for that element. restricted NLTE NLTE calculations are performed only in the line formation, without feedback eﬀects on the model atmosphere.

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

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

S/N Signal-to-noise ratio.

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

SGB Sub-giant branch, stars just past the TOP. species An element in a speciﬁc ionization stage.

71 Logarithmic increase/decrease (dex) 0.01 0.05 0.10 0.30 0.50 Linear increase (%) 2.3 12 26 100 216 Linear 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 an analysis, which are usually possible to shift without affecting the results of the analyses. This because spectroscopic mechanisms may give only differentially consistent rather than absolute values. the cosmological lithium-problem A disagreement between the calculated primordial lithium abundance from standard big bang nucleosynthesis calibrated by WMAP data, and the abundance inferred from the so-called Spite plateau in metal-poor stars. The disagreement is on the order of 0.3 − 0.5 dex, corresponding to a factor 2 − 3.

TOP Turnoff-point, where a dwarf has nearly exhausted the available hydrogen in the core, and rapidly transforms into a red giant across the SGB and bRGB stages over a billion years (cf. roughly ten billion years on the main sequence). traditional stellar models Stellar models which do not treat the effects of atomic diffusion.

72 Nomenclature

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

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

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

+200e Another temperature scale, following +200 but with surface gravities spectroscopically determined using the ionization equilibrium – thus e for equilibrium. 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 photosphere, defined from the relation to luminosity and radius in the Stefan- 2 4 Boltzmann law, L = 4πR σTeff.

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 relative 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 a medium.

73 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 Korn et 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.