Exploring Non-Standard Stellar Physics with Lithium Depletion

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Garrett Evan Somers

Graduate Program in Astronomy

The Ohio State University 2016

Dissertation Committee: Professor Marc Pinsonneault, Advisor Professor Jennifer Johnson Professor Donald Terndrup Copyright by

Garrett Evan Somers

2016 Abstract

For over a century, astronomers have been building physical models to study the internal constitution of stars. These eﬀorts have culminated in the so-called

“standard stellar model” (SSM), the computational bedrock upon which the modern theory of stellar evolution stands. SSMs have been extremely successful at describing many observational properties of stars, including the intricate morphology of the color-magnitude diagrams of star clusters, the observed mass-radius relationship on the hydrogen-burning main sequence, and the detailed interior structure of the Sun.

Nonetheless, there remain several outstanding features of the galactic stellar pattern that SSMs fail to predict, and it is these discrepancies which point to interesting physical processes occurring in stars that have heretofore been neglected.

One signiﬁcant tool for uncovering deﬁciencies in the SSM is the abundance of the light element lithium. Li is rapidly destroyed deep inside stellar atmospheres, duly depleting the observed abundance at the surface, and in principle revealing detailed information about the temperatures, mixing processes, and physical conditions of the interiors of stars. SSMs make strong predictions for the destruction of Li in solar-type stars, namely that a mass-dependent depletion pattern is imprinted on the pre-main sequence, and persists in an unaltered state for the

ii entirety of the main sequence. However, these predictions are strongly contradicted by numerous observations, which implying unequal rates of Li burning at ﬁxed mass and composition during the pre-main sequence, and variable, long-timescale

Li destruction during the main sequence. Unraveling the underlying processes driving this non-standard Li depletion will inform our understanding of the physical phenomena which are ignored in our standard picture, but are inﬂuential to the structure and evolution of stars.

In this dissertation, I report on my progress reconciling SSMs with the open cluster Li pattern by accounting for the eﬀects of metallicity, rotation, magnetism, starspots, and radius inﬂation. First, I demonstrate that SSMs accurately predict the Li abundances of solar analogs at the zero-age main sequence within theoretical uncertainties, and that rotational mixing and core-envelope coupling can accurately predict the main sequence depletion pattern. Next, I suggest that the dispersion in

Li abundances at fixed effective temperature observed at the zero-age main sequence results from the inflated radii of rapidly rotating objects. Then, I support this suggestion by calculating rotating evolutionary models, and showing that radius inflation can indeed suppress the Li destruction driven by rotation. Finally, I suggest that starspots may be the underlying physical mechanism connecting rotation with radius inflation. I incorporate a treatment of starspots in my evolutionary models, and find good agreement with the observed Li pattern.

iii Dedication

To my wonderful wife, Jenna.

iv Acknowledgments

First, I would like to express my immense gratitude to my thesis adviser Marc

Pinsonneault. His contagious enthusiasm and enduring dedication to educating graduate students made my professional life at Ohio State a joy every single day.

I cannot recall a single instance during my entire graduate career when he did not make time to answer questions I had, or to react (positively!) to a crazy idea that popped into my head, or to just chat about this and that, for which I will always be grateful. It goes without saying that the research presented in this thesis would not have happened without him, and any future success I may have in this ﬁeld will in part be a product of the lessons (implicit and explicit) he has taught me about how to be a researcher.

Although each faculty member contributes in their own way to the wonderful environment in this department, I would like to particularly recognize three other professors with whom I have worked during the last ﬁve years. First, I have had the pleasure of working with Don Terndrup in my last two years. Amusingly, our collaboration began with a random conversation outside a McDonald’s at lunch time, and developed into weekly meetings and always entertaining conversations. I would also like to thank him for his eﬀorts in writing letters of recommendation for

v me during job season. Although never formally a collaborator, I would also like to thank Jennifer Johnson for the innumerable scientific (and otherwise) conversations occurring during our weekly stars meetings, and of course for authoring letters of recommendation for me. Finally, I would like to thank Smita Mathur for serving as my adviser during my first year. Having never formally done research before graduate school, I was forced to learn an awful lot about astronomy during my first project at Ohio State, and her guidance helped steer my aimless wanderings to an ultimately successful first crack at it.

Finally, I would like to thank my family for their unending love and support during my (so far) 27 years on Earth. My interest in math science grew when I was very young, and their encouragement kept it alight all these years. Look where I am now! And last, but most certainly not least, thank you to my amazing best friend and wife Jenna, for her love, her support, her companionship, and so much more.

vi Vita

November 16, 1988 ...... Born – Boston, MA, USA

University of California, San Diego B.S. Applied Mathematics 2011 ...... B.S. Physics

2011 – 2015 ...... Graduate Teaching and Research Associate,

The Ohio State University 2015 – 2016 ...... Presidential Fellow, The Ohio State University

Publications

Research Publications

1. G. Somers, S. Mathur, P. Martini, C.J. Grier, L. Ferrarese, “Discovery of a Large Population of Ultraluminous X-ray Sources in the Bulge-less Galaxies NGC 337 and ESO 501-23”, ApJ, 777, 7, (2013).

2. G. Somers, and M.H. Pinsonneault, “A Tale of Two Anomalies: Depletion, Dispersion, and the Connection Between the Stellar Lithium Spread and Inﬂated Radii on the Pre-Main Sequence”, ApJ, 790, 72, (2014).

3. G. Somers, and M.H. Pinsonneault, “Rotation, inﬂation, and lithium in the Pleiades”, MNRAS, 449, 4131S, (2015).

4. G. Somers, and M.H. Pinsonneault, “Older and Colder: The Impact of Starspots on Pre-Main Sequence Stellar Evolution”, ApJ, 807, 174, (2015).

5. G. Somers, and M.H. Pinsonneault, “ Lithium depletion is a strong test

vii of core-envelope recoupling”, arXiv 1606.00004, (2016, in press).

6. L.M. Rebull, et al. (eighteen co-authors including G. Somers), “Rotation in the Pleiades with K2: I. Data and First Results”, arXiv 1606.00052, (2016, in press).

7. L.M. Rebull, et al. (seventeen co-authors including G. Somers), “Rotation in the Pleiades with K2: II. Multi-Period Stars”, arXiv 1606.00055, (2016, in press).

8. J.R. Stauﬀer, et al. (seventeen co-authors including G. Somers), “Rotation in the Pleiades With K2: III. Speculations on Origins and Evolution”, arXiv 1606.00057, (2016, in press).

9. A.M. Boesgaard, et al. (ﬁve co-authors including G. Somers), “Boron Abundances Across the ”Li-Be Dip” in the Hyades”, arXiv 1606.07117, (2016, in press).

Fields of Study

Major Field: Astronomy

viii Table of Contents

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vii

List of Tables ...... xiii

List of Figures ...... xiv

Chapter 1: Introduction ...... 1 1.1 Standard Stellar Models: Strengths and Limitations ...... 1 1.2 LithiuminAstrophysics ...... 3 1.3 Standard Predictions of Stellar Lithium Depletion ...... 4 1.4 TheObservedClusterPattern ...... 6 1.5 What Mechanisms Could Be Responsible? ...... 8 1.6 ScopeofthisDissertation ...... 11

Chapter 2: A De-biased Measure of Main Sequence Lithium Depletion 19 2.1 Background ...... 19 2.2 Methods...... 22 2.2.1 Cluster Selection and Parameters ...... 22 2.2.2 AbundanceAnalysis ...... 24 2.2.3 StellarModels...... 27

ix 2.3 SourcesofError...... 28 2.3.1 MetallicityErrors...... 29 2.3.2 Theoretical Systematics ...... 30 2.4 Calibrating Standard Stellar Model Physics ...... 32 2.5 Results...... 35 2.5.1 BenchmarkClusters ...... 37 2.5.2 AdditionalClusters...... 44 2.6 Empirical Test of the Metallicity Dependence of Lithium Depletion . 53 2.6.1 Does Metallicity Impact Pre-MS Lithium Depletion? . . . . . 53 2.6.2 Lithium as a Tracer of Cluster Metallicity ...... 58 2.7 Summary ...... 60

Chapter 3: Suppressed Convection and the Radius Anomaly as the Origin of the Lithium Dispersion in Young Cool Dwarfs? ...... 77 3.1 Background ...... 77 3.2 TheRadiusAnomaly...... 80 3.3 LithiumandRadiusinthePleiades ...... 82 3.4 AdditionalClusters...... 85 3.5 Consequences of the Radius Anomaly in Young Clusters ...... 90 3.6 Summary ...... 93

Chapter 4: Rotation, Inflation, and Lithium in the Pleiades ...... 101 4.1 Background ...... 101 4.2 Methods...... 107 4.2.1 Standard model calibration and radius inflation ...... 108 4.2.2 Rotation and angular momentum evolution ...... 111 4.2.3 Pleiadesdata ...... 115 4.3 Theimpactofrotationandinflation ...... 115 4.3.1 Rotation...... 116 4.3.2 Inflation ...... 120

x 4.3.3 Rotation and inflation ...... 122 4.4 Sleuthing the rotation–inflation connection ...... 124 4.4.1 CaseI:noinflation ...... 126 4.4.2 Case II: uniform inflation ...... 127 4.4.3 Case III: uncorrelated inflation ...... 129 4.4.4 Case IV: correlated inflation ...... 130 4.5 Discussion...... 135 4.5.1 The mechanism of inflation ...... 135 4.5.2 Testing the inflation paradigm ...... 140 4.6 Summary ...... 143

Chapter 5: The Impact of Starspots on Pre-Main-Sequence Stellar Evolution ...... 155 5.1 Background ...... 155 5.2 Methods...... 158 5.2.1 Standardstellarmodels ...... 158 5.2.2 Spots...... 159 5.3 Effectonstellarradii ...... 165 5.3.1 Thepre-mainsequence ...... 165 5.3.2 The zero-age main sequence ...... 167 5.3.3 Comparisonwithdata ...... 169 5.4 Effect on inferred masses, ages, and colors ...... 170 5.4.1 Spot effects in the Hertzsprung-Russell diagram ...... 172 5.4.2 Spot effects in the color-magnitude diagram ...... 175 5.5 Effect on lithium depletion and cluster dating ...... 179 5.6 Discussion...... 184 5.6.1 A radius dispersion during the pre-main sequence ...... 184 5.6.2 Literaturecomparison ...... 188 5.6.3 Observational and physical concerns ...... 191 5.7 Summary ...... 196

xi Chapter 6: Lithium Depletion is a Strong Test of Core-Envelope Recoupling ...... 210 6.1 Introduction...... 210 6.2 Methods...... 216 6.2.1 Rotationdata ...... 216 6.2.2 Lithiumdata ...... 218 6.2.3 Stellarmodels...... 219 6.3 Rotation and mixing in stellar models ...... 221 6.3.1 Rotation in limiting case models ...... 222 6.3.2 Rotation in hybrid models ...... 224 6.3.3 Mixing...... 225 6.4 Mycalibrationprocedure...... 227 6.5 Results...... 231

6.5.1 Constraining D0 withrotationdata ...... 231 6.5.2 Lithiumdepletion...... 235 6.5.3 The timescales of core-envelope recoupling ...... 238 6.6 Discussion...... 242 6.6.1 The solar rotation proﬁle ...... 242 6.6.2 WhyisM67sodiﬀerent?...... 243 6.7 SummaryandConclusions ...... 248

References ...... 262

xii List of Tables

Table1.1 BenchmarkClusterData ...... 18 Table2.1 Pre-MSSystematicErrors...... 75 Table 2.2 Benchmark Cluster Lithium Anomalies ...... 76 Table3.1 AdditionalClusterData...... 100 Table 5.1 Mass and Age Correction Factors ...... 207 Table 5.1 Mass and Age Correction Factors ...... 208 Table 5.1 Mass and Age Correction Factors ...... 209

xiii List of Figures

Figure 1.1 Dependence of theoretical lithium depletion on metallicity. . . 16 Figure 1.2 Comparing benchmark cluster lithium data to standard predictions...... 17 Figure 2.1 Systematic errors in lithium and temperature derivations. . . . 64 Figure 2.2 Dependence of theoretical lithium depletion on model physics. 65 Figure 2.3 Influence of systematics on detrended cluster lithium...... 66 Figure 2.4 Fitting the Pleiades lithium pattern...... 67 Figure 2.5 Corrected standard model lithium depletion for benchmark clusters...... 68 Figure 2.6 Final lithium anomaly measurements for benchmark clusters. . 69 Figure 2.7 Comparison with literature lithium depletion measurements. . 70 Figure 2.8 Timescales of lithium depletion and dispersion...... 71 Figure 2.9 Impact of metallicity on lithium data at 2 Gyr...... 72 Figure 2.10 Impact of metallicity on lithium data at 600 Myr...... 73 Figure 2.11 Lithium as a metallicity tracer in old clusters...... 74 Figure 3.1 Impact of radius inflation on lithium depletion at 120 Myr. . . 95 Figure 3.2 Cluster parameter errors and inflated radius lithium depletion (1/3)...... 96 Figure 3.3 Cluster parameter errors and inflated radius lithium depletion (2/3)...... 97 Figure 3.4 Cluster parameter errors and inflated radius lithium depletion (3/3)...... 98 Figure 3.5 Impact of radius inflation on luminosity and temperature.. . . 99 Figure 4.1 Rotation and lithium in the Pleiades...... 146 Figure 4.2 Radius anomaly versus mixing length...... 147

xiv Figure 4.3 Influence of rotation on lithium depletion...... 148 Figure 4.4 Influence of radius inflation on lithium depletion...... 149 Figure 4.5 Influence of rotation and inflation on lithium tracks...... 150 Figure 4.6 Influence of rotation and inflation on lithium depletion patterns. 151 Figure 4.7 Testing limiting rotation/inflation connections...... 152 Figure 4.8 Testing the best-case rotation/inflation correlation...... 153 Figure 4.9 Varying the strength of the rotation/inflation correlation. . . . 154 Figure 5.1 Impact of starspots on radii, luminosities, and Teff...... 199 Figure 5.2 Impact of starspots on the mass-radius relationship...... 200 Figure 5.3 Impact of starspots on the masses and ages of young stars. . . 201 Figure 5.4 Impact of starspots on young cluster color-magnitude diagrams. 202 Figure 5.5 The effet of spot temperature on color-magnitude diagrams. . 203 Figure 5.6 Impact of starspots on lithium depletion cluster ages...... 204 Figure 5.7 How starspots inhibit the depletion of lithium...... 205 Figure 5.8 The impact of spot depth on radius inflation...... 206 Figure 6.1 Angular momentum evolution of limiting case and hybrid models.251 Figure 6.2 Lithium depletion of limiting case and hybrid models...... 252 Figure 6.3 Illustration of my calibration procedure...... 253 Figure 6.4 Constraining my models with open cluster rotation data. . . . 254 Figure 6.5 Constraining my lithium depletion models with solar mass clusterdata...... 255 Figure 6.6 Solar-calibrated hybrid models in different mass bins...... 256 Figure 6.7 The best fit hybrid models for each mass bin...... 257 Figure 6.8 My best fit core-envelope coupling timescales...... 258 Figure 6.9 Rotation profiles of solar-mass, solar-age hybrid models. . . . . 259 Figure 6.10 Double Gaussian fit of the h Persei rotation distribution. . . . 260 Figure 6.11 Forward models of clusters with different rapid rotator fractions. 261

xv Chapter 1: Introduction

1.1. Standard Stellar Models: Strengths and Limitations

Stars are one of the principal building blocks of the Universe, and are central to countless sub-ﬁelds in astronomy. Stars are the birthplaces of planets, and their ages provide the primary constraint on the timescales of planet formation and evolution.

Stars are also a key tracer of the assembly and chemical enrichment history of the

Milky Way. Finally, stars are interesting laboratories for fundamental and exotic physics, playing host to such phenomena as dynamos, pulsational instabilities, angular momentum transport and loss, and starspots. Given their paramount importance, astronomers have labored for over a century to construct physicals model which can accurately and consistently explain the observable properties of stars. These eﬀorts have culminated in the current bedrock of stellar theory, the standard stellar model (SSM).

Standard stellar models are a speciﬁc formalism for predicting the structure and evolution of stars (see, e.g., Serenelli 2016), and are implemented today in many complementary computer codes. These codes generally begin with the four equations governing the structure of stars, namely mass and energy conservation,

1 energy transport, and hydrostatic equilibrium. In order to properly specify the variables within these equations, a series of prescriptions for stellar microphysics are applied. These include treatments for temperature- and composition-dependent opacities, equations of state, nuclear reaction networks, surface boundary conditions, a treatment of convection (usually done in the mixing-length formalism; e.g.

B¨ohm-Vitense 1958), and more. The solutions to the equations of stellar structure are then allowed to evolve by solving for the relevant time-dependent physical phenomena, like Hayashi contraction, radiative emission from the surface, and core nuclear burning. SSMs have proven extremely successful for predicting the behavior of stars, and have some notable triumphs such as correctly reproducing the observable features of cluster color-magnitude diagrams, accurately predicting the solar neutrino ﬂux, and localizing the base of the solar convection zone.

However, it is important to note that SSMs are designed primarily to predict the properties of stars similar to the Sun, and thus lose their predictive power in certain regimes of stellar behavior. This is because SSMs do not attempt to treat certain physical phenomena that are unimportant for the Sun, but which may exert inﬂuence for other objects. Examples include rotation, which is a small eﬀect in the

Sun but which is thought to alter the lifetimes and surface chemical abundances of more rapidly-rotating stars (e.g. Pinsonneault 1997), and starspots, which can inhibit energy transport and alter the boundary conditions at the surface, leading to a signiﬁcant structural response (e.g. Spruit 1982a). Given the importance of these

2 phenomena, particularly for young stars which rotate rapidly and are often heavily spotted, it is important to consider their impact on stellar model predictions. My dissertation focuses on this very topic. In the coming chapters, I exploring the impact of non-standard-stellar-model physics on the structure and evolution of stars, using a sensitive diagnostic, lithium.

1.2. Lithium in Astrophysics

Lithium (Li) has a wide variety of uses in contemporary astrophysics. First, the Li abundance of old, metal-poor stars is a strong constraint on theories of Big Bang

Nucleosynthesis (BBN). This is because Li was created in trace amounts during the

Big Bang (e.g. Boesgaard & Steigman 1985), and thus the primordial Li content of the Universe should be reﬂected in the original abundance patterns of the earliest generations of stars. Second, Li is a tracer of the evolution of Milky Way chemistry.

This is because various exotic galactic phenomena generate Li through 3He +4 He

7 and Be + e− reactions, and enrich the interstellar medium. This gradual enrichment is apparent in our own Sun, whose primordial abundance of A(Li) = 3.3 is a factor of

4(!) larger than the BBN prediction of A(Li) = 2.7. The dominant site of Li creation in the galaxy is still debated (see Prantzos 2012 for a review), but likely includes contributions from classical novae (Starrﬁeld et al. 1978), asymptotic giant branch nucleosynthesis (Sackmann & Boothroyd 1992), cosmic-ray spallation (Reeves 1970), and white dwarf-white dwarf mergers (Longland et al. 2012).

3 Finally, the evolving Li abundances of solar-metallicity stars is a signature and test of their structure and the physics operating in their interiors. This is because

Li is eﬃciently destroyed by the 7Li(p,γ)8Be nuclear reaction in stellar material

6 hotter than ∼ 2.5 × 10 K (referred to as TLB). This temperature is reached at the base of the convection zone of solar-type stars during the pre-main sequence, leading to a gradual depletion of the initial surface abundance, but not during the main sequence. Due to the extreme sensitivity of this destruction rate to the local temperature (∝ T 20), the evolving Li abundances of stars are a sensitive probe of their internal temperature structure. In order to put this property of Li to use, we must ﬁrst review the predicted pattern of Li abundances, and compare the results to the empirical Li pattern. Any discrepancies between theory and observations in this exercise might be a product of non-standard physical phenomena.

1.3. Standard Predictions of Stellar Lithium Depletion

Strong theoretical predictions of the astration of lithium have existed now for half a century (e.g. Iben 1965), but as I will show, remain woefully inadequate for describing the observed cluster pattern. To understand why the standard predictions for lithium depletion fail in some regimes, we must ﬁrst understand the standard picture of stellar lithium destruction.

Standard predictions for Li depletion derive from the standard stellar model, and are well established. The sole mixing mechanism in SSMs is convection, so the

4 surface Li abundance of a star is predicted to decrease only in stars with convective envelopes (M <1.3M 1), and only when the temperature at the base of their surface ⊙ convection zones, which I call TBCZ , approaches or surpasses TLB. SSMs predict that this occurs only the pre-MS, but not on the MS. Pre-MS stars have deep convective envelopes, and heat up as they contract, causing TBCZ to eventually surpasses TLB and inducing Li depletion. In fully convective stars (FCSs; M < ⊙

0.35M ), Li is completely destroyed in only a few Myr once TLB is reached. This ⊙ occurs earlier for higher mass objects, creating a boundary between FCSs which have depleted Li, and lower mass FCSs which remain Li-rich. The location of this boundary is age dependent, permitting its use as a method for dating clusters; this is called the lithium depletion boundary (LDB) technique (see Section 5.5). Stars less massive than ∼0.06M never reach TLB in their interior, and so retain their ⊙ initial Li abundance forever.

For stars more massive than 0.35M , the convective envelope begins to retreat ⊙ on the late pre-MS; this causes TBCZ to once again cool below TLB, and terminates

Li depletion. Lower mass stars take longer to reach TLB, but remain in the burning phase for longer. This results in greater depletion factors in these objects. Li also burns more rapidly in metal-rich stars, as the higher resulting opacity deepens the convective envelope, increasing TBCZ . The result is a strongly mass and metallicity

1In this dissertation, I restrict myself only to stars below this mass threshold.

5 dependent lithium depletion pattern (LDP) on the zero-age main sequence with no dispersion at ﬁxed mass.

Examples of SSM lithium depletion patterns for diﬀerent metallicities can be seen in Fig. 1.1. For a solar metallicity cluster, Li has been almost completely destroyed in stars <0.6M by the ZAMS, but only about ∼ 20% has been destroyed ∼ ⊙ for 1.3M . Once on the MS, the Li depletion zone resides inside the radiative core, ⊙ and SSMs predict no more mixing-related Li depletion until the main sequence turn-oﬀ. The LDP will continue to change on Gyr timescales, but this is not a result of mixing, and is instead due to the evolving temperature of main sequence stars, and gravitational settling in late F and early G dwarfs.

The expected history of Li depletion in solar-type stars can therefore be summarized as follows. They are born with an abundance similar to the proto-solar value (A(Li) = 3.3), undergo an epoch of depletion during the pre-MS where they burn an amount which depends on their mass and metallicity, cease Li depletion once they reach the main sequence, and retain their ZAMS abundance up until they leave the main sequence.

1.4. The Observed Cluster Pattern

Having established the basic expectations for lithium depletion in solar-type stars, we now examine the open cluster Li pattern. Open clusters are the ideal laboratory for validating lithium depletion predictions, as they provide stars with a range

6 of masses at ﬁxed age and composition. Many open cluster Li data sets exist in the literature, but the general observational picture can be accurately represented by three well-studied clusters. Fig. 1.2 shows empirical Li data for the Pleiades,

Hyades, and M67, alongside SSM predictions calculated for their respective cluster parameters (solid black lines; see Section 2.2.1-2.2.3 and Table 1.1 for details).

As illustrated by this ﬁgure, there are several inconsistencies between standard predictions and observed cluster patterns:

(i) The median abundance is over-predicted by a few tenths of a dex above

6100K, and under-predicted by a few tenths of dex to greater than an order-of- magnitude below 6100K. While the general trend of greater depletion in cooler stars is accurately predicted, this SSM marginally fails to predict the median of solar analogs, and catastrophically fails to predict the median of cool stars.

(ii) There is a signiﬁcant scatter (∼ 2 dex) in surface Li abundances in the

Pleiades below 5500K. This implies that additional physical parameters, which can vary between equal-Teﬀ stars, aﬀect pre-MS depletion by orders-of-magnitude in this temperature regime. Furthermore, the fastest rotating cool Pleiads are on average the most Li-rich stars at their respective temperatures (Soderblom et al. 1993a).

This strongly implies a connection between rotation and Li depletion.

(iii) Fig. 1.2 shows a strong temporal evolution of the median Li abundance at all masses. From left to right, SSMs ﬁrst under-predict, then over-predict, then

7 greatly over-predict the median pattern at 100 Myr, 600 Myr, and 4 Gyr. This implies that mechanisms other than convection are able to mix stellar material on the MS. Furthermore, the rate of depletion decreases at advanced ages (Sestito &

Randich 2005).

(iv) By the age of M67, a large Li scatter at ﬁxed Teﬀ has developed in solar analogs. This scatter is not present in the Pleiades, and so likely develops during the MS. This demonstrates that the Li abundance of a given star depends on factors other than just mass, age, and metallicity. Additional physical parameters that vary between equal-mass stars must induce this relative depletion.

These four points strongly contradict the standard picture of lithium burning, and likely point to additional pieces of stellar physics that have been neglected in

SSMs. In this dissertation, I will present possible sources of each discrepancy.

1.5. What Mechanisms Could Be Responsible?

The inconsistencies described in Section 1.4 imply that, in contrast to standard theory, the Li patterns of main sequence open clusters are the product of two distinct processes: a pre-MS process that imparts a strongly mass-dependent median trend with a variable width, and a longer timescale processes on the main sequence that causes the median and dispersion to evolve with time.

8 Little theoretical eﬀort has gone into explaining the inhibited depletion developing on the pre-MS (i), and its apparent dependence on rapid rotation (ii).

However, some mechanisms have been proposed, such as suppressed shear mixing due to eﬃcient core-envelope recouping in rapid rotators (Bouvier 2008), episodic accretion during the proto-stellar phase altering the interior temperature (Baraﬀe

& Chabrier 2010), and star-disk coupling inducing strong internal mixing shears

(Eggenberger et al. 2012). However, I believe that the most promising explanation was the one proposed by King et al. (2010): increased stellar radii impacting depletion eﬃciency. If a star’s radius is inﬂated on the pre-main sequence, its interior will be cooled by a corresponding amount. This can slow the rate of Li destruction by proton capture, and lead to a higher abundance on the zero-age main sequence.

In Chapters 3-5, I lay out an argument supporting this hypothesis, culminating in what I believe to be the underlying cause of pre-main sequence radius inflation, starspots. Spots are the visual manifestation of concentrated magnetic fields in the convective envelopes of late-type stars. These fields inhibit the flow of energy in specific regions near the surface, inducing a structural response (Gough & Tayler

1966; Spruit 1982a,b). One of the structural consequences is an increased radius, resulting from the altered boundary conditions at the surface. The photospheric pressure is higher for the hotter ambient photosphere than it would be for a model with a surface temperature of a given Teﬀ at ﬁxed radius. The model expands as a result in order to reach equilibrium (P ∝ M 2/R4 in homology), which lies at higher

9 radius and lower surface pressure. Consequently, the interior of the star is cooled, thus suppressing the rate of Li destruction. More heavily spotted stars are more strongly cooled on the pre-main sequence, ultimately resulting in a dispersion at

ﬁxed Teﬀ on the zero-age main sequence.

On the other hand, main sequence Li depletion been discussed for several decades (e.g. Herbig 1965; Zappala 1972; Balachandran 1995; Pinsonneault 1997;

Sestito & Randich 2005), but the mechanism, or mechanisms, responsible have yet to be deﬁnitively established. Suspects include mixing driven by rotation and angular momentum (AM) loss (Pinsonneault et al. 1989; Zahn 1992; Chaboyer, Demarque &

Pinsonneault 1995), mixing driven by internal gravity waves (Press 1981; Montalb´an

& Schatzman 2000), dilution of the envelope through mass loss (Weymann & Sears

1965; Swenson & Faulkner 1992), and microscopic diﬀusion (Richer & Michaud

1993).

In this dissertation, we explore a particularly promising explanation, rotational mixing (e.g. Pinsonneault et al. 1990; Pinsonneault, Deliyannis & Demarque 1992;

Zahn 1992; Chaboyer, Demarque & Pinsonneault 1995; Sestito & Randich 2005; Li et al. 2014). Theory predicts that chemical materials can be exchanged between layers in the interiors of rotating stars due to several hydrodynamic processes, such as

Eddington–Sweet circulation (Eddington 1929; Sweet 1950; Zahn 1992) and secular shear instabilities (e.g. Zahn 1974). Through these mechanisms, Li-depleted material from the deep interior can be mixed into the convective envelope, diluting the surface

10 abundance and decreasing A(Li) below the standard model predictions. Rotational mixing has been invoked to explain the progressive destruction of Li observed in main sequence open clusters (described in (iii)), because it operates over time-scales analogous to main sequence lifetimes (Pinsonneault et al. 1990). It also naturally predicts the development of Li dispersions (described in (iv)), found in some (but not all) old clusters (e.g. Sestito & Randich 2005), since depletion rates are determined by the non-uniform angular momentum (AM) histories of individual stars (e.g. Irwin

& Bouvier 2009). Finally, rotational mixing decreases in eﬃciency at late ages due to stellar spin down, permitting the observed plateauing of Li abundances seen for old clusters (Sestito & Randich 2005). To investigate this mechanism, I will make a precision measurement of the debiased rate of main sequence Li depletion to serve as a ﬁgure of merit for mixing calculations (Chapter 2). Then, I will perform rotational mixing calculations, with both traditional limiting cases of angular momentum transport, and a new method which allows for stronger core-envelope coupling than permitted by pure hydrodynamics, and compare the results to data (Chapter 6).

1.6. Scope of this Dissertation

For my dissertation, I have published three papers (Somers & Pinsonneault 2014,

2015a,b) and have a fourth accepted (Somers & Pinsonneault 2016), which attempt to understand and quantify the source of these discrepancies. I present these works in ﬁve chapters. The ﬁrst and last deal with Li depletion on the main sequence and

11 the impact of composition on Li burning. The middle three concern the spread in lithium abundances that develops during the pre-MS.

In Chapter 2, I first demonstrate that SSMs accurately predict the Li abundances of solar analogs at the zero-age main sequence (ZAMS) within theoretical uncertainties. I then measure the rate of main sequence Li depletion by removing the [Fe/H]-dependent ZAMS Li pattern from three well-studied clusters, and comparing the detrended data. Main sequence depletion is found to be mass dependent, in the sense of more depletion at low mass. A dispersion in Li abundance at fixed Teff is nearly universal, and sets in by ∼200 Myr. I discuss mass and age dispersion trends, and the pattern is mixed. Next, I argue that metallicity impacts the ZAMS Li pattern, in agreement with theoretical expectations but contrary to the findings of some previous studies, and suggest Li as a test of cluster metallicity.

In Chapter 3, I argue that a radius dispersion in stars of fixed mass and age, during the epoch of pre-MS Li destruction, is responsible for the spread in Li abundances and the correlation between rotation and Li in young cool stars, most well known in the Pleiades. To support this notion, I calculate stellar models, inflated to match observed radius anomalies in magnetically active systems, and the resulting range of Li abundances reproduces the observed patterns of both the Pleiades and several other young open clusters. I discuss ramifications for pre-MS evolutionary tracks and age measurements of young clusters, and suggest an observational test.

12 In Chapter 4, I expand on these calculations by including the effects of rotational mixing during the pre-MS. I demonstrate that pre-MS radius inflation can efficiently suppress lithium destruction by rotationally induced mixing in evolutionary models, and that the net effect of inflation and rotational mixing is a pattern where rotation correlates with lithium abundance for M < M , and ∗ ⊙ anti-correlates with lithium abundance for M > M , similar to the empirical trend ∗ ⊙ in the Pleiades. Next, I adopt different prescriptions for the dependence of inflation on rotation, and compare their predictions to the Pleiades lithium/rotation pattern.

I find that if a connection between rotation and radius inflation exists, then the important qualitative features of this pattern naturally and generically emerge in my models. This is the first consistent physical model to date that explains the

Li–rotation correlation in the Pleiades. I discuss plausible mechanisms for inducing this correlation and suggest an observational test using granulation.

In Chapter 5, I assess whether the physical mechanism driving this correlation between rotation rate and radius inflation could be starspots. I find that heavily spotted stellar models of mass 0.1-1.2M are inflated by up to 10% during the ⊙ pre-MS, and up to 4% and 9% for fully- and partially-convective stars at the zero-age

MS, consistent with measurements from active eclipsing binary systems. Spots similarly decrease stellar luminosity and Teﬀ , causing isochrone-derived masses to be under-estimated by up to a factor of 2×, and ages to be under-estimated by a factor of 2-10×, at 3 Myr. Consequently, pre-MS clusters and their active stars are

13 systematically older and more massive than often reported. Cluster ages derived with the lithium depletion boundary technique are erroneously young by ∼ 15% and 10% at 30 and 100 Myr respectively, if 50% spotted stars are interpreted with unspotted models. Finally, lithium depletion is suppressed in spotted stars with radiative cores, leading to a ﬁxed-temperature lithium dispersion on the main sequence if a range of spot properties are present on the pre-MS. Such dispersions are large enough to explain Li abundance spreads seen in young open clusters, and imply a range of radii at ﬁxed mass and age during the pre-MS Li burning epoch.

By extension, this implies that mass, composition, and age do not uniquely specify the HR diagram location of pre-MS stars.

Finally, in Chapter 6, I perform a new test of rotational mixing, a prime candidate to explain the Li from the photospheres of cool stars during the main sequence. Previous mixing calculations have relied primarily on treatments of angular momentum transport in stellar interiors incompatible with solar and stellar data, in the sense that they overestimate internal differential rotation. Instead, recent studies suggest that stars are strongly differentially rotating at young ages, but approach solid body rotation during their main sequence lifetimes. I modify my rotating stellar evolution code to include an additional source of angular momentum transport, a necessary ingredient for explaining the open cluster rotation pattern, and examine the consequences for mixing. I confirm that core-envelope recoupling is required to explain the evolution of the mean rotation pattern along the main

14 sequence, and demonstrate that it also provides a more accurate description of the

Li depletion pattern seen in open clusters. Recoupling produces a characteristic pattern of efficient mixing at early ages and little mixing at late ages, thus predicting a flattening of Li depletion at a few Gyr, in agreement with the observed late-time evolution. Using Li abundances, I argue that the timescale for core-envelope recoupling during the main sequence is a strong function of mass. I discuss implications of this finding for stellar physics, including the viability of gravity waves and magnetic fields as agents of angular momentum transport. I also raise the possibility of intrinsic differences in initial conditions in star clusters, using M67 as an example.

15 1.30M 3.5 ⊙ 1.20M ⊙1.10M ⊙ 1.00M ⊙ A(Li) 0.90M ⊙ initial 0.80M ⊙ 3.0

0.70M ⊙ 2.5

2.0 Metal Poor 1.5 A(Li) Metal 1.0 Rich

0.5

[Fe/H] = 0.00 0.60M ⊙ 0.0 [Fe/H] = ± 0.10 [Fe/H] = ± 0.20 −0.5 [Fe/H] = ± 0.30 7000 6500 6000 5500 5000 4500 4000 Effective Temperature (K)

Fig. 1.1.— Dependence of theoretical lithium depletion on metallicity. SSM Li predictions at the ZAMS for a variety of metallicities, which illustrate the strong dependence of pre-MS Li depletion on composition. The lowest metallicity pattern is the top-most curve, showing that increased metallicity drastically increases the Li depletion rate at early stages of stellar evolution. X and Z values are calculated with respect to a Grevesse & Sauval (1998) proto-solar abundance. Dashed lines display curves of constant mass, and the dotted line represents the initial lithium abundance.

16 3.5 3.5 Pleiades Hyades M67 3.0 125 Myr 625 Myr 3.9 Gyr 3.0

2.5 2.5

2.0 2.0

A(Li) 1.5 1.5

1.0 1.0

0.5 0.5

0.0 0.0 6500 6000 5500 5000 4500 6500 6000 5500 5000 4500 6500 6000 5500 5000 4500 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 1.2.— Comparing benchmark cluster lithium data to standard predictions. Comparing benchmark cluster lithium data to standard predictions.Lithium data for the Pleiades, Hyades, and M67 are shown alongside standard stellar model lithium patterns calculated for their respective cluster parameters (solid black lines). Arrow denote upper limits.

17 Cluster # EW(Li)1 B-V Age [Fe/H] E[B-V] Parameter Sources stars Source Source (error) (error) (Age, [Fe/H], E[B-V])

Pleiades 115 (1) (1) 125(5) Myr +0.03(0.02) Variable2 (2), (3), (1) Hyades 65 (4) (5) 625(25) Myr +0.135(0.005) 0.01 (6), (7), (8) M67 56 (9) (10) 3.9(0.6) Gyr +0.01(0.03) 0.041 (11), (11), (12) 18 1Sources: (1) Soderblom et al. 1993a; (2) Stauffer, Schultz & Kirkpatrick 1998; (3) Soderblom et al. 2009; (4) Thorburn et al. 1993; (5) Johnson & Knuckles 1955; (6) Perryman et al. 1998; (7) Cummings et al. 2012; (8) Lyng˚aet al. 1985; (9) Pasquini et al. 2008; (10) Montgomery et al. 1993; (11) Castro et al. 2011; (12) Taylor 2007. 2Due to significant differential extinction in the Pleiades, Soderblom et al. (1993a) de-reddened each star individually. I adopt their values.

Table 1.1. Benchmark Cluster Data Chapter 2: A De-biased Measure of Main Sequence Lithium Depletion

2.1. Background

In Chapter 1, I outlined four key discrepancies between standard predictions for lithium depletion and the observed cluster pattern. In particular, I noted that Li is destroyed efficiently on the MS, over a timescale comparable to the lifetimes of solar-type stars. In this chapter, I will study the depletion of Li on the MS, produce an empirical measure of this depletion by applying corrections to the data which account for metallicity effects, and study the timescales of the emergence of dispersions in Li abundance at fixed mass on the MS.

Accurate depletion measurements can be used both to anchor mixing calculations on empirical data and to place constraints on proposed mixing mechanisms. In all cases, I will refer to the diﬀerence between SSM predictions, and the empirical abundance of stars, as the lithium anomaly. While previous authors have measured main sequence Li depletion by comparing the abundances of diﬀerent-aged main sequence clusters (e.g. Sestito & Randich 2005), these studies have generally not accounted for one crucial element: higher metallicity stars are

19 expected to deplete greater amounts of Li during the pre-MS. This effect can severely bias comparisons in absolute space, as the ZAMS abundance at a given Teff may differ between clusters by up to an order of magnitude (Section 2.3.1). To address this, I present a novel method in Section 2.4 for quantifying the lithium anomaly that develops on the MS. I will argue that the main sequence anomaly signal can be isolated from an empirical main sequence Li pattern by subtracting a SSM

LDP from the data. This removes the relative, [Fe/H]-dependent pre-MS depletion signal, leaving behind the depletion induced by non-standard main sequence mixing.

Although some authors have claimed that this metallicity eﬀect is not supported by observational evidence, I present a case in Section 2.6.1 that composition is indeed central in shaping ZAMS Li patterns.

The eﬃcacy of this method hinges on the quality of SSM LDP predictions, which we know from (i) in Section 1.3 can be poor. Therefore, I must ﬁrst reconcile my theoretical predictions with the data. To do this, I explore the possibility that errors in model input physics account for this discrepancy. The extreme sensitivity of the rate of Li burning to the surrounding temperature (∝ T 20; Bildsten et al.

1997) implies that minute changes in TBCZ on the pre-MS may have large effects on the magnitude of Li depletion predicted in SSMs. TBCZ in a stellar model may be affected by the assumed physics, so in order to validate this method, I first address the following question: can SSMs accurately predict the magnitude of pre-MS Li depletion within the errors of my adopted input physics? If this is so, I can use

20 empirical data to calibrate my SSMs, and produce accurate predictions of pre-MS Li depletion (Section 2.4).

This chapter is organized as follows. In Section 2.2.1, I describe the open cluster Li data sets used for my analysis. Section 2.2.2 describes the equivalent width and photometric data I use to infer stellar parameters of my benchmark clusters, and the abundance analysis I employ. Section 2.2.3 describes the stellar evolution code used to generate theoretical LDPs, and enumerates the physics in my ﬁducial calculations. I then present an exploration of uncertainties inherent to the detrending process. In Section 2.3.1, I quantify the impact of [Fe/H] errors in my benchmark clusters, and conclude that extremely well-constrained composition is necessary to accurately predict the pre-MS signal. In Section 2.3.2, I perform a systematic study of the eﬀects of various physical inputs on SSM LDPs. I then constrain the input physics in my models with the empirical Pleiades Li pattern, and describe my method of cluster detrending in Section 2.4. In Section 2.5.1, I measure the lithium anomaly in the benchmark clusters, and compare the results to previous calculations. This measurement will ultimately anchor the mixing included in my rotating stellar models. I then revisit the data collected in Sestito & Randich

(2005) with my detrending methodology in Section 2.5.2, and explore the evolution of both median Li abundances and Li dispersion along the MS. In Section 2.6.1, I argue that composition is an important factor in shaping ZAMS LDPs and validate

21 my anomaly measurements, and in Section 2.6.2 suggest Li as a precision test of the metallicity of clusters.

2.2. Methods

2.2.1. Cluster Selection and Parameters

Cluster Li data will serve several purposes in this paper. First, I will use three benchmark clusters to precisely measure the rate of main sequence Li depletion. The

Pleiades will be used to calibrated the physics in my theoretical models, by requiring that its SSM Li pattern agree with its empirical Li pattern at the ZAMS, and the relative abundances of the Hyades and M67 will be used to infer the main sequence lithium anomaly, by comparing their detrended patterns to those expected from the

Pleiades calibration, corrected for their metallicities. Second, I will reanalyze the data of Sestito & Randich (2005), and examine the timescales of Li depletion and dispersion (Section 2.5.2). Third, I will argue that metallicity plays an important role in pre-MS Li destruction by comparing similar-aged clusters of dissimilar composition (Section 2.6.1).

Benchmark Clusters

The literature hosts a wealth of Li data for FGK dwarfs in open clusters (Sestito

& Randich 2005; Torres et al. 2006; Sacco et al. 2007; Prisinzano & Randich 2007;

Pasquini et al. 2008; Randich et al. 2009; Jeﬀries et al. 2009; Anthony-Twarog et al.

22 2009; Cargile et al. 2010; Balachandran et al. 2011; Cummings et al. 2012; Pace et al. 2012; Fran¸cois et al. 2013). Li depletion calculations are exquisitely sensitive to composition, so I restrict my potential choices to clusters with small [Fe/H] errors

(< 0.05 dex) to minimize uncertainties. This excludes all but the most well-studied clusters. Furthermore, large Li data sets are required to minimize errors resulting from shot noise, because dispersion is a ubiquitous feature.

With these considerations in mind, I select the Pleiades, Hyades, and M67 as my benchmark clusters. These are well-suited for this investigation because they are exceptionally well studied, thus minimizing errors associated with photometry, extinction, binarity, membership, and most importantly, composition (Table 1.1).

The Pleiades is 125 ± 5 Myr old (Stauﬀer, Schultz & Kirkpatrick 1998), making it my near ZAMS cluster, The Hyades is 625 ± 25 Myr old (Perryman et al. 1998), and M67 is 3.9 ± 0.6 Gyr old (Castro et al. 2011). This level of temporal coverage allows us to characterize the relative strengths of early and late-time mixing.

Additional Clusters

In Section 2.5.2, I will revisit the clusters examined by Sestito & Randich (2005) through the use of my detrending analysis (see their table 1 for details). For both the benchmark clusters and those considered in Section 2.6.1, I will adopt the data sets described in this text. For the rest, I adopt the photometry and Li equivalent widths (EWs) reported by Sestito & Randich (2005), and use the analysis techniques

23 described in Section 2.2.2. I adopt the cluster reddening, ages, and Fe abundances reported by Sestito & Randich (2005), except for the following cases, where I have substituted higher resolution metal abundances: [Fe/H] = -0.03 ± 0.04 for IC4665

(Shen et al. 2005), [Fe/H] = 0.00 ± 0.01 for IC 2602 (D’Orazi & Randich 2009),

[Fe/H] = -0.01 ± 0.02 for IC 2391 (D’Orazi & Randich 2009), [Fe/H] = 0.04 ± 0.02 for Blanco 1 (Ford et al. 2005), [Fe/H] = 0.01 ± 0.07 for NGC 2516 (Terndrup et al.

2002), and [Fe/H] = 0.03 ± 0.02 for NGC 6475 (Villanova et al. 2009).

2.2.2. Abundance Analysis

For each benchmark cluster, I drew λ6707.8 Li I EWs, and photometric BV measurements, from various literature sources (see Table 1.1). The Pleiades EWs and photometry come from Soderblom et al. (1993a). Hyades EWs come from

Thorburn et al. (1993) and Hyades photometry comes from Johnson & Knuckles

(1955). M67 EWs come from Pasquini et al. (2008) and M67 photometry comes from Montgomery et al. (1993). To maximize the internal consistency of my data sets, I did not merge multiple catalogs of Li EWs or photometry. I applied the reddening corrections referenced in Table 1.1 to these data, calculated eﬀective temperatures using the BV polynomial ﬁt of Casagrande et al. (2010), and derived

A(Li)s with the curves of growth (CoG) of Soderblom et al. (1993a). These CoG are valid between 4000K, where total depletion on the pre-MS occurs, and 6500K, where signiﬁcant additional non-standard mixing occurs on short time scales (the

24 lithium dip Boesgaard & Tripicco 1986; Balachandran 1995), so I discard stars that lie outside these bounds. This does not aﬀect my conclusions, as my main concern is solar analogs.

The Li absorption line suﬀers from blending with a nearby Fe I line located at

6707.4A.˚ Although the resolution of the Hyades spectra of Thorburn et al. (1993) was high enough to directly remove the blend, the resolution of the Pleiades and M67 spectra were not. The authors therefore removed the blend contribution using the method suggested by Soderblom et al. (1993a): the Fe I contribution is calculated as EW(λ6707.4 Fe I) = 20(B − V ) − 3 mA,˚ and subtracted from the measured EW.

They estimate that this relation is accurate to 3 − 5mA,˚ signiﬁcantly smaller than the errors on the Pleiades EWs, but comparable to the M67 EWs. This may impact the inferred abundances of stars with low EWs, for which blends are naturally harder to remove. However, the [Fe/H] of M67 is similar to the cluster this relation was calibrated on (the Pleiades), so the errors are likely on the low end of the range.

Furthermore, systematic oﬀsets should not aﬀect comparisons of Pleiades and M67 stars, given that their EWs were obtained with the same deblending process. Finally, this may introduce minor systematic errors between these data and the Hyades sample (see Thorburn et al. 1993 for a discussion).

To estimate the accuracy of these abundances, I compare Hyades A(Li)s derived from the Casagrande et al. (2010) Teﬀ scale to abundances derived using temperatures obtained with the An et al. (2007) Teﬀ scale. This comparison is shown

25 in the left panel of Fig. 2.1. Both derivations used the CoG of Soderblom et al.

(1993a), to isolate the effect of Teff systematics on my final results. The abundances agreed to better than 0.05 dex for all stars. Errors in Teff do not have a strong impact on LDPs, because Teff and A(Li) are correlated such that Teff errors move stars diagonally along the pattern (Fig. 2.1; right panel). Though systematic offsets may affect the relative amounts of stars in each Teff and A(Li) bin, the median of the pattern is largely unaffected. Next, I combine the Hyades Teff and Li EWs presented in Steinhauer (2003) with the Soderblom et al. (1993a) CoG, and compared the resulting abundances to those presented by Steinhauer (2003), who used his own

CoG. This is illustrated in the center panel of Fig. 2.1. There is good agreement for stars with A(Li) ∼> 2, but the derived abundances for Li-poor stars diﬀer by up to

0.22 dex between the CoGs. This is a potentially signiﬁcant systematic error source, so I perform the analysis with both CoGs and compare the results in Section 2.5.1.

Finally, I compare the derived data for each cluster to alternative measurements from the literature. The comparison data were drawn from Margheim (2007) for the Pleiades, Takeda et al. (2013) for the Hyades, and Sestito & Randich (2005) for M67. The Pleiades and Hyades sets are quite similar to the alternative choices, particularly in the 5500-6100K range, where I desire the cleanest sample (Section

2.4). Equivalent width measurements diﬀer more in the case of M67, but this is largely due to the improved data quality of Pasquini et al. (2008) relative to that of

Jones et al. (1999). The former authors achieved a higher average S/N (∼ 95) than

26 the latter (∼ 80), so I believe that my chosen sample is superior. I caution that the

Pasquini et al. (2008) sample has relatively low resolution spectra (R ∼ 17,000), and as a result the quoted upper limits for their non-detections may be optimisticly low.

I defer discussion of this particular issue to Section 2.5.

2.2.3. Stellar Models

To calculate the SSM predictions of the LDPs of open clusters, I use the Yale

Rotating Evolution Code (see Pinsonneault et al. 1989 for a discussion of the mechanics of the code). I adopt a Grevesse & Sauval (1998) proto-solar metal abundance (Z/X = 0.025293; this is larger than the current solar surface abundance

Z/X = 0.02292 due to gravitational settling), and choose the solar hydrogen mass fraction and the mixing length (α) such that a solar mass model reproduces the solar luminosity and radius at 4.57 Gyr. The calibrated values are α = 1.88269,

X = 0.71304, the helium mass fraction Y = 0.26882, and metal mass fraction Z

= 0.018035 for [Fe/H] = 0.0. My models use the 2006 OPAL equation of state

(Rogers, Swenson & Iglesias 1996; Rogers & Nayfonov 2002), atmospheric initial conditions from Kurucz (1979), high temperature opacities from the opacity project

(Mendoza et al. 2007), low temperature opacities from Ferguson et al. (2005), and the 7Li(p, α)α cross section of (Lamia et al. 2012). In each of my models, I assume a initial Li abundance equal to the proto-solar abundance: A(Li) = 3.31 (Grevesse

& Sauval 1998). I return to this assumption when evaluating the accuracy of

27 my measurements, but for now mention that because Li depletion is logarithmic, dA(Li)/dt does not depend on the abundance.

To obtain the chemical mixture for arbitrary [Fe/H], I adopt a big bang helium mass fraction YBB = 0.2484 (Cyburt, Fields & Olive 2004), and assume a linear evolution of Y with metals:

∆Y Y = Y + Z (2.1) BB ∆Z

∆Y /∆Z is solved for by calculating the slope between the big bang mixture (Y ,Z)=

(0.2484,0.0), and my calibrated solar mixture (Y ,Z) = (0.26882,0.018035). I derive

∆Y /∆Z = 1.13, consistent with recent estimates (Casagrande et al. 2007). I use this method to create abundance mixtures for the adopted [Fe/H] of the clusters in this study, and evolve forward models of mass 0.5-1.3M , in steps of 0.05M , for each. ⊙ ⊙

2.3. Sources of Error

I first consider sources of error that could affect the accuracy of my ZAMS Li predictions. These fall into two categories: errors affecting the relative predictions between cluster LDPs, and errors affecting the absolute predictions of all cluster

LDPs. The relative error budget is dominated by uncertainties in cluster [Fe/H], which shift the predicted LDPs of clusters relative to one another, impacting the inferred lithium anomaly (Section 2.3.1). This error has been minimized in my analysis through the selection of clusters with exquisitely measured [Fe/H].

Absolute errors are dominated by uncertainties in the physics adopted in my

28 models, the physics of Li burning, and the proto-solar abundance, which aﬀect the

LDP predictions of all clusters simultaneously. Although these uncertainties are systematic, they can impact the relative predictions of clusters, and thus the inferred anomaly. I describe the eﬀect of these errors in Section 2.3.2, and account for their impact in Section 2.4.

2.3.1. Metallicity Errors

ZAMS LDPs are extremely sensitive to composition. This is demonstrated by Fig.

1.1, which shows several Pleiades-age patterns calculated with a range of [Fe/H]s.

As can be seen, Li abundances vary greatly at fixed Teff and fixed mass (dashed lines) depending on the composition. More metal-rich clusters are progressively more depleted, and in some cases, a deviation of 0.1 dex in [Fe/H] produces a deviation ∼> 1 dex in A(Li)! This strong dependence results from a deeper CZ in metal-rich stars, which increases TBCZ and drastically increases the burning rate.

This eﬀect is much stronger in lower mass stars, suggesting that the greatest stability to [Fe/H] uncertainties can be found in and above the solar regime. By contrast, the rate of depletion on the main sequence is insensitive to composition. Regardless of metallicity, TBCZ on the main sequence is much less than TLB, suggesting that the rate of Li destruction cannot depend on the thermal properties of the envelope during this epoch. Therefore, once the [Fe/H]-dependent pre-MS signal is removed, comparisons between main sequence LDPs become stable to metallicity errors.

29 2.3.2. Theoretical Systematics

Lithium depletion is extremely sensitive to the adopted stellar physics (D’Antona &

Mazzitelli 1994; Tognelli et al. 2012). There are several physical inputs that could potentially impact Li predictions, the most important of which I list in Table 2.1.

To estimate the theoretical errors associated with these components on the pre-MS,

I adopted the SSM prediction for the Pleiades as a fiducial model. I then varied each source of uncertainty in turn, computed the resulting LDP, and compared it to my fiducial pattern. The results can be seen in the left column of Fig. 2.2. The top panel shows the differences between the fiducial pattern and each alternative pattern as function of Teff , and the bottom panel shows the quadrature sum of the uncertainties, with the width of the error band enumerated at periodic intervals. I treat the sum of systematic differences between inputs, such as distinct equation of state tables or opacity calculations, as effective 2σ errors.

My theoretical errors are asymmetric. This is largely because changes in the solar heavy element mixture systematically reduce Li burning; adopting the Asplund et al. (2009) abundance ratios has an effect analogous to reducing [Fe/H] by ∼ 0.1 dex. The equation of state is the second largest effect, because it sets the relationship between T and other physical parameters of the model. Other significant effects include the choice of model atmosphere, though this is a large effect only for stars

7 with main sequence Teﬀ < 5000K, and the Li(p, α)α cross-section, which produces symmetrical LDPs about the ﬁducial choice. There is a larger dynamic range in the

30 model uncertainties for cool stars than for hot stars, similar to errors induced by metallicity uncertainties, because of their lengthier pre-MS burning phase.

I repeated the above exercise to investigate the impact of theoretical uncertainties on the minor SSM depletion occurring on the MS. To do this, I evolved my ﬁducial and alternate stellar models from the age of the Pleiades to 2 Gyrs, and measured the additional discrepancy that develops during this time period. This can be see in the right column of Fig. 2.2. The impact on hot stars is negligible.

The base of the CZ above 5500K is so cool during this period that small changes to the thermal structure of the envelope do not result in substantial changes to the rate of Li destruction. Minor changes are seen in cool stars, whose CZ bases are still somewhat warm, but at a much lower level than the uncertainties developing on the pre-MS. This demonstrates that once a depletion pattern has been corrected for theoretical errors arising on the pre-MS, the inferred main sequence depletion is stable.

Fig. 2.3 shows data from my benchmark clusters detrended with respect to the bottom and top of the pre-MS systematic error band, and the ﬁducial model.

While [Fe/H] errors move individual clusters with respect to one another, theoretical uncertainties move the clusters up and down in tandem. Uncertainties inherent to my models are predominately an absolute bias level in the magnitude of Li depletion, rather than a relative error between the clusters. Nevertheless, the inferred difference between clusters can change by a few tenths of a dex at fixed Teff depending on the

31 choice of systematics, because more metal-rich clusters have a wider systematic error band. For instance, the anomaly measurement at 5800K between the Pleiades and

Hyades is diﬀerent in the left and right panels of Fig. 2.3. This demonstrates the need for empirical guidance in the selection of physical inputs.

2.4. Calibrating Standard Stellar Model Physics

I have shown that Li depletion is sensitive to a large number of physical inputs.

In the absence of compelling information about which inputs are most in error, I cannot pinpoint which parameter, or parameters, should be changed to reconcile my

ﬁducial theoretical models with nature. However, since main sequence depletion is insensitive to physics in SSMs, and because there is great theoretical freedom in the

ZAMS pattern, it is reasonable to adopt an empirical ﬁt to the absolute depletion of a ZAMS cluster. This gives us an ad hoc calibration of the ensemble physics in my models. Once this has been done for one cluster, SSMs predict the age and metallicity scalings required to obtain calibrated predictions for other clusters. This scaling is sensitive to errors in metallicity, but if the composition of a cluster is reliable, analysis on detrended warm cluster stars will not be biased by signiﬁcant failures of standard models.

Lithium patterns of young clusters, such as the Pleiades, are expected to closely mimic the true ZAMS LDP, as they have undergone minimal main sequence depletion. I can therefore use the empirical Pleiades pattern to guide my selection

32 of the theoretical LDP that best reflects the true ZAMS distribution. Before I can do this, I must choose a suitable Teff range to use for this calibration. As described in Section 2.3, errors induced by both observational and theoretical uncertainties are more significant for cool stars than for hot stars. I therefore restrict my model calibration to stars with Teff > 5500K. At this Teff , the width of the theoretical error band is still large (∆A(Li) ∼ 0.7 dex), but the random errors due to uncertainties in [Fe/H] are small (∆A(Li) ∼< 0.2 dex; see Fig. 1.1). This is beneficial, since I have great leverage in calibrating the input physics due to the large theoretical uncertainties, and relatively small relative errors due to [Fe/H] uncertainties.

Additionally, I do not include stars with Teﬀ > 6100K in this analysis, because early mixing could be impacting their abundances (Margheim 2007). My ﬁnal analysis regime is 5500-6100K.

I then interpolate to each theoretically allowed ZAMS LDP inside the 2σ error bounds in the top left panel of Fig. 2.2, and determine how well it matches the data by computing the median absolute deviation (MAD) of the empirical Pleiades Li distribution around it. I then select the LDP that produces the best goodness-of-ﬁt within the calibration regime. The resulting function is shown in Fig. 2.4. Here, the black and red lines are the same as in Fig. 2.2, and the blue line represents the best

ﬁt model. The empirical Pleiades data follow this curve well, with the exception of a few outliers, which may reﬂect true scatter. I adopt this as my calibrated Pleiades

33 SSM, and as the function I will use to detrend out pre-MS depletion from Pleiades data.

The calibrated patterns for other clusters will lie the same fractional distance between their fiducial LDPs and their upper theoretical error envelopes as the calibrated Pleiades SSM does. To calculate these models, I derive a scale factor from the Pleiades system by dividing the absolute depletion of the detrending function by the absolute depletion of the fiducial model. The scale factor is nearly invariant within the Teff analysis domain, allowing us to approximate it as a constant.

Scaling the Hyades ﬁducial model by this value produced a close approximation to a detrending function generated by fully calculating the theoretical errors for

Hyades parameters and interpolating to the same location as in the Pleiades. This demonstrates that this method can be used to generate systematics-corrected LDPs for any given set of cluster parameters. My calibrated models for the benchmark clusters are shown as solid black lines in Fig. 2.5, alongside the ﬁducial SSMs, which are represented by dashed lines. As can be seen, the tension between the median depletion predicted at ZAMS and solar analogs in the Pleiades has been resolved.

This ﬁgure also demonstrates that the lower envelope of the cool star dispersion in the Pleiades is well approximated by SSMs. This implies that whatever mechanism is inducing this spread does so by suppressing Li depletion in rapid rotators, and not by inducing additional Li depletion in slow rotators (see Chapter 3).

34 I note that the metal content of my models has been determined by scaling the proto-solar abundance by the measured [Fe/H] of each cluster. In actuality, [Fe/H]s are reported relative to the solar photospheric abundance. The true metallicity of stars during the pre-MS was larger than their current abundances by a factor equal to the amount of gravitational settling occurring during their lifetime. This is a mass and age dependent eﬀect, in the sense that older and more massive stars undergo more settling. The magnitude of the eﬀect for solar analogs in M67 is ∆[Fe/H] =

-0.06, leading to a 0.04 dex change in the predicted pre-MS Li depletion. This effect is smaller for lower mass stars, because settling is negligible when the CZ is deep, and smaller for higher mass stars as well, because the rate of Li depletion is insensitive to composition in this regime. The corresponding effect is significantly lower for the Hyades, since it is younger by a factor of 8, and non-existent for the Pleiades, since I have calibrated my models on empirical data for this cluster. Given the small magnitude of the effect, and the complex, and generally unknown, dependence of settling on mass and age, I do not include this effect in my calculations.

2.5. Results

Surface Li destruction proceeds rapidly on the pre-MS, and much slower thereafter.

To empirically measure the rate of Li depletion on the MS, I must accurately predict the amount of depletion that occurs for stars on the pre-MS, so it can be subtracted from their present day abundances. However, the amount of Li destruction occurring

35 on the pre-MS in SSMs is highly sensitive to errors in both the assumed metallicity of the cluster and the physical inputs in my models. The former can be controlled by considering clusters with well known composition, so I have selected three well-studied clusters for a precision measurement. The latter can be controlled by selecting the input physics which best reproduces the observed Li pattern of a

ZAMS cluster, which I do in an ensemble fashion in Section 2.4. Once my physics is calibrated, the Li destruction occurring on the pre-MS can be detrended out of an empirical Li pattern, thus isolating the depletion occurring on the MS. The rate of main sequence Li depletion can then be inferred by comparing the average Li anomalies of diﬀerent aged clusters.

In this section, I use this methodology to obtain the main sequence Li anomaly for several clusters. First, I will detrend the Hyades and M67, and infer from their anomaly patterns the rate of main sequence Li depletion, as a function of mass and age. I ﬁnd a strong mass trend in the rate of main sequence depletion in both clusters, and conﬁrm that the average rate of depletion decreases at advanced ages.

I also discuss a few caveats to this method, including the effects of the initial Li abundances of clusters, and the possibility of cosmic variance differentially impacting cluster LDPs, but conclude that my results are robust. I then detrend and analyze a large sample of open clusters, previously studied by Sestito & Randich (2005), to explore the timescales of main sequence depletion and dispersion. I find that the rate of depletion of solar analogs is unchanged by the transformation to anomaly

36 space, but the mass dependence of depletion is stronger in absolute space. Finally,

I measure the Li dispersion at fixed Teff in each cluster, and explore the timescales of its development. I find that dispersion sets in early in many clusters, and can increase, decrease, or remain steady over time depending on the temperature bin.

Furthermore, dispersion is not a simple function of age, suggesting cosmic variance in cluster initial conditions.

2.5.1. Benchmark Clusters

The Lithium Anomaly

My detrended benchmark clusters are shown in Fig. 2.6. Here, I have divided my analysis regime into three bins of 200K width. This provides us sufficiently low shot noise to calculate the median Li depletion without introducing too much error due to the Teff dependence of the pattern. I also include two additional Teff regimes that were excluded in Section 2.4: 6300-6100K and 5350-5150K. Although these temperature ranges were not suitable for calibrating my models, the anomaly remains a useful measure of mixing. The light red, light blue, and light purple stars represent the average Teff and the median lithium anomaly of the Pleiades, Hyades, and M67 within each bin (illustrated by the vertical dashed lines). I have ignored the Teff range 5500-5350K, due to a gap in the Hyades data in this regime.

Between the Pleiades and the Hyades, I ﬁnd the following lithium anomalies, where the ﬁrst quoted error is due to Poisson noise, and the second is due to [Fe/H]

37 uncertainties: 0.020 ± 0.124 ± 0.018 dex at 6200K, 0.183 ± 0.057 ± 0.018 dex at

6000K, 0.353 ± 0.055 ± 0.025 dex at 5800K, 0.984 ± 0.111 ± 0.021 dex at 5600K, and ∼> 1.636 dex at 5250K. Between the Pleiades and M67, I ﬁnd anomalies of 0.471

± 0.094 ± 0.025 at 6200K, 0.901 ± 0.082 ± 0.041 dex at 6000K, 1.730 ± 0.128 ±

0.058 dex at 5800K, and ∼> 2.436 dex at 5600K. There is no anomaly for the 5250K bin here, since no M67 Li data exists in the literature in this range. The Hyades anomaly at 5250K and the M67 anomaly at 5600K are upper limits, due to the preponderance of upper limits in this bin. These measurements are collected in

Table 2.2.

In Section 2.2.2, I compared the Soderblom et al. (1993a) CoG used in this paper with the Steinhauer (2003) CoG, and found substantial differences in final abundance for some stars. To evaluate the impact of this uncertainty on my final answer, I compare my answers to anomaly values derived using the Steinhauer

(2003) CoG. These are also shown in Table 2.2. For all but the coolest Teﬀ bin, the relative anomalies between the Pleiades and Hyades agree with my original values at ∼ 1σ or better. Between the Pleiades and M67, the values agree at ∼ 1.5σ or better. This increases my conﬁdence that these results are robust. The uncertainties are larger for M67, since fractional errors increase when EWs are small. This could be particularly troublesome for the 5600K bin if the reported upper limits are inaccurate; for instance, Onehag¨ et al. (2011) measured A(Li) = 1.3 for M67-1194, a star for which I derive an upper limit of A(Li) = 0.53 using EWs from Pasquini

38 et al. (2008). While such errors would not impact the median in bins dominated by detections, they could revise the upper limit in the 5600K bin upwards by ∼ 0.8 dex.

Finally, the lower limits derived with Steinhauer (2003) are ∼0.2 dex lower for the

5250K bin, reﬂecting the decreasing quality of curves of growth for low-abundance stars. This error is not too worrisome, as 0.2 dex represents <15% of the total anomaly at this temperature.

An important conclusion about main sequence Li depletion in FGK dwarfs can be drawn from this plot: low mass stars deplete Li more rapidly than high mass stars on the MS. While this eﬀect has been seen previously in absolute space (e.g. Sestito

& Randich 2005), my work confirms that the effect persists when the additional depletion suffered by low-mass stars on the pre-MS has been removed. This result holds regardless of the choice of theoretical bias and CoG. This provides a stringent constraint on mechanisms seeking to explain the main sequence lithium anomaly in open clusters. A second conclusion I can draw from this plot is that the average rate of main sequence depletion is higher for the Hyades than for M67. This can be seen in Fig. 2.6 by the ratio of the anomaly at the age of M67 to the anomaly at the age of the Hyades. If the depletion rate were constant for these two clusters, this ratio should be ∼ 7.6, equal to the ratio of time the clusters have spent on the MS.

However, in the 5800K and 6000K bins, the ratio is 4.8 and 5.1 respectively. This depletion plateau has been seen before, but I have shown that the result persists even when diﬀerential metallicity eﬀects are accounted for.

39 The Li content of the interstellar medium has increased over time (Spite & Spite

1982), and a correlation between cluster Fe abundance and initial Li abundance has also been reported (Cummings 2011), so it is possible in principle that my benchmark clusters began their lives with diﬀerent Li abundances. If true, this would serve to strengthen my conclusions. The initial M67 Li content is likely close to solar, since it is nearly equal to the Sun in both age and metallicity. The initial Pleiades Li abundance has also been measured to be near solar (e.g. Cummings 2011). However, the Hyades may have been born with a higher Li abundance than I have assumed.

If this is correct, the true magnitude of main sequence depletion for the Hyades will be greater than I have measured. This would reduce the diﬀerence between the

Hyades and M67 medians, decrease the ratio described above, and therefore increase the tension between my measurement and the putative expectations of constant logarithmic depletion. Furthermore, changing the assumed initial abundance of a cluster moves each of its members up and down in tandem, and so will not change the mass-dependent pattern revealed through detrending. I therefore believe that shifts in the initial cluster abundance could impact the precision of my anomaly measurements, but will not impact either of the conclusions stated above. A rigorous evaluation of the Hyades initial cluster abundances can be undertaken by analyzing the Li content of stars blue-ward of the Li gap, as they will have suﬀered minimal pre-MS and main sequence depletion.

40 It is plausible that additional cluster parameters could impact the Li pattern, jeopardizing the generality of my measured anomalies. For example, rotation is expected to drive deep mixing flows through meridional circulation and shear instabilities, and induce non-standard Li depletion during the main sequence as a result (e.g. Pinsonneault 1997, and refs. therein). The rate of depletion increases with faster rotation, so clusters with different rotation distributions will eventually develop different LDPs. Environment could therefore impact the Li depletion properties of clusters.

The early (0-10 Myr) rotation evolution of a star is dictated by its circumstellar disk, which locks magnetically to the star and eﬃciently drains AM from the envelope (e.g. Koenigl 1991; Rebull et al. 2006). If a T Tauri star has a close interaction with another cluster member, its circumstellar disk may be subject to early disruption through the interaction. This will truncate the timescale of the disk-locking phase, and the star will retain the AM it would have otherwise lost, appearing as a rapid rotator at the ZAMS. Such interactions are more likely in dense stellar environments, so this process could induce a correlation between the number density of a cluster at birth, and the fraction of rapidly rotating stars. Assuming a connection between rapid rotation and main sequence Li depletion, dense clusters would be expected to host a commensurately large fraction of Li-poor stars in the solar regime. This would impact both the width of the Li distribution, and the median anomaly. Although this explanation is qualitatively sensible, Bouvier et al.

41 (1997) found that the rotation rates of primaries in binary systems are statistically indistinguishable from rotation rates of single stars in the Pleiades. This suggests that physics local to the star sets rotation rates, and not environmental factors.

Literature Comparison

I now compare my results to Sestito & Randich (2005), who calculated the average

Li abundance in three Teﬀ bins at a variety of ages along the MS. It should be noted that there are two important diﬀerences between my analysis and that of

Sestito & Randich (2005). First, I measured relative abundances in lithium anomaly space, whereas Sestito & Randich (2005) worked in absolute abundance space. The practical effect is that Sestito & Randich (2005) did not correct for differential depletion on the pre-MS due to composition differences. Second, I analyzed only one cluster at a time, whereas Sestito & Randich (2005) grouped several similarly-aged clusters together and calculated an ensemble average. Their method tends to wash out both random errors in Teff and A(Li), and the effect of different metallicities.

With these precautions, I compare results in Fig. 2.7. In the top panel, the red points are my lithium anomaly measurements for the Hyades, and the blue points represent the diﬀerence between the bins containing the Hyades and the bins containing the Pleiades in Sestito & Randich (2005). The measurements are generally consistent with one another. Between 5700K and 6200K, I measure a marginally smaller Hyades depletion than Sestito & Randich (2005). This is because

42 the Hyades is a metal-rich cluster, and thus less main sequence depletion is inferred when pre-MS eﬀects are considered. However, my measurement is lower than the

Sestito & Randich (2005) measurement at 5600K. Given the uncertainties inherent to CoG analysis in low abundance regimes, this may reflect differences in the abundance derivation rather than a true difference in measurements. Alternatively, given the steepness of the LDP in this Teff range, this difference may be due to random errors in my sample. More stars populate the warm side than the cool side of this bin, biasing my answer towards greater depletion.

The red points in the bottom panel of Fig. 2.7 represent my lithium anomaly measurements for M67. Sestito & Randich (2005) did not bin M67 with other clusters, but instead measured the average of the M67 upper and lower envelopes separately. I therefore show the range in which the global average of their sample could reside. My measurement at 6000K is in good agreement with Sestito & Randich

(2005). This is not surprising, since M67 and Pleiades are very similar in [Fe/H], and thus pre-MS corrections are minimal. My 5800K and 5600K measurement are large compared to Sestito & Randich (2005), but this is likely due to the diﬀerent data samples used in the two studies. The sample of Sestito & Randich (2005) is dominated by Li measurements from Jones et al. (1999), which had a lower detection threshold than Pasquini et al. (2008), used by this work. I thus ﬁnd a lower average at 5800K, and set a stricter upper limit at 5600K.

43 2.5.2. Additional Clusters

Minimal measurement errors have made the Hyades and M67 optimal for calibrating and testing mixing calculations, but two temporal points do not provide a complete time line of Li evolution. I therefore apply an analysis similar to Section 2.5.1 to the full sample of open clusters described in Sestito & Randich (2005). They assembled all substantial FGK dwarf Li data sets available prior to 2005, and applied a uniform abundance analysis to examine the timescales of Li evolution. The results were the conﬁrmation of a mass trend in the rate of main sequence depletion, and the identiﬁcation of four stages of depletion: depletion on the pre-MS, a stall near the

ZAMS, depletion on the MS, and a plateau at late ages. Given the considerations put forth in this paper, I wish to determine if these qualitative and quantitative results are altered by the transformation to anomaly space. To this end, I reanalyze with my methods the data for each cluster described in Section 2.2.1, with the exception of my benchmark clusters, where I reuse the data sets employed in Section

2.5.1. The clusters NGC 2264 and NGC 2547 are excluded, since the former is too young for substantial pre-MS Li depletion to have occurred, and the latter lacks a quality [Fe/H] measurement.

Evolution of the Median

I ﬁrst compared my results with those of Sestito & Randich (2005) in absolute space.

The data for each cluster were divided into the three Teﬀ regions they considered

44 (5600 ± 100K, 5900 ± 150K, and 6200 ± 150K), and the median of each bin was calculated. I then combined the data from similar-aged clusters in the fashion of Sestito & Randich (2005) (see their table 3), to maximize the validity of the comparison. Unsurprisingly, my findings are consistent with theirs. A decline in median abundance as a function of age is present in each Teff bin, and the rate of depletion increases with decreasing mass. Modest differences between my points and theirs are present, due to the difference between mean and median statistics, and my choice of different data sets for some clusters. Nevertheless, this confirms the similarity of the two analysis processes.

Next, I detrended each cluster, using the machinery of Section 2.2.3 and the systematic corrections of Section 2.4, and binned the data as described above. I display each cluster as a single data point instead of applying age bins, allowing for a visual impression of the intrinsic scatter of cluster medians about the mean trend. Finally, I rejected bins with less than 3 members, and plotted the data in the ﬁrst three panels of the left column of Fig. 2.8. Black points represent the median anomaly of each cluster in the quoted bins, and the error bars represent the quadrature sum of the standard error of the median ( MAD ), and the uncertainty of √N the median due to [Fe/H] errors. I also plot the data of Sestito & Randich (2005) in red, with the relevant scale displayed on the right axis. Finally, I binned and detrended data in the Teﬀ range 5300 ± 100K, and plotted the data in the bottom left panel.

45 As evinced by this ﬁgure, cooler stars possess a higher average depletion rate over their lifetimes in both absolute and anomaly space, in agreement with the

findings of Section 2.5.1 and Sestito & Randich (2005). In order to quantify this effect, I calculate the best fit power law for the data in each bin up to 2 Gyr, when the subdivision of M67 in the Sestito & Randich (2005) data and upper limits begin to complicate the regression process. The red line in each panel shows the best fit for the Sestito & Randich (2005) data, and the slope is reported in the lower left corner. The 6200K bin depletes slower than the 5900K bin by a factor of 1.9, and slower than the 5600K bin by a factor of 2.9. This trend is preserved in anomaly space, but the mass dependence is found to be marginally steeper: depletion in the the detrended 6200K bin is consistent with the absolute 6200K bin, but is slower by a factor of 2.4 compared to the 5900K bin, and slower by a factor of 4.4 compared to the 5600K bin. The bottom left panel of Fig. 2.8 shows that this trend continues beyond the Teff range considered by Sestito & Randich (2005); the depletion rate at

5300K is a factor of 6.2 greater than in the warmest bin. The anomalies up to ∼150

Myr are positive in this bin, reflecting the over-abundance of cool stars relative to theory (see Chapter 3), and drop sharply after this age. Therefore, the rate of main sequence depletion after this age is likely higher than reflected by the best fit power law slope.

Sestito & Randich (2005) argued that abundances in each Teﬀ bin converge to a plateau after about 2 Gyr. However, this late-time plateau behavior does not

46 satisfactorily describe the median of M67 in the three warmest bins. While the upper envelope of M67 shares the same average abundance as the 2 Gyr bin in the

Sestito & Randich (2005) data, the large dispersion present in this cluster causes the median abundance to be lower by several tenths of a dex. This can clearly be seen in my data in the 5900K bin. Furthermore, the NGC 188 data (5 Gyr bin in the Sestito

& Randich (2005) data) are in fact subgiants, and were therefore hotter on the main sequence than currently observed. As a result, they should not be compared with cooler dwarfs. The behavior of Li at late ages is altered when this evolution eﬀect is accounted for. Instead of ceasing, as the red points in the 5900K bin imply, depletion appears to continue as a power-law function of time, indicated by the black points in the 6200K bin. Whether a dispersion similar to that seen in M67 arrises in NGC

188 is unclear, because the abundances of cool stars in this cluster remain unprobed.

Our understanding of Li depletion beyond the solar age is clearly incomplete, and requires further observational work.

Evolution of the Dispersion

Some proposed main sequence mixing mechanisms naturally predict a range of depletion rates at ﬁxed mass, and thus a variable dispersion as a function of time

(i.e. rotational mixing), while others predict that all stars of a given mass deplete

Li at equal rates on the MS, implying a ﬁxed dispersion (i.e. gravity wave mixing).

The evolution of the distribution of Li abundances at fixed Teff therefore provides an additional constraint on main sequence mixing. This quantity is difficult to measure

47 in absolute space, as strong mass trends impart a large difference in average A(Li) at the high and low ends of cool Teff bins. This introduces significant scatter about the median abundance of the bin, rendering this measurement impossible without excellent number statistics. However, this issue can be resolved by transforming the data to Li anomaly space. In this plane, the intrinsic scatter due to the presence of pre-MS mass trends is removed, and a robust measurement of dispersion is possible.

I calculated the standard deviation of each Teff bin from Section 2.5.2, and plot the results in the right column of Fig. 2.8. The black circles represent the measured dispersions, and the diameter of the circle reflects the sample size: the smallest circles have 3 members, and the largest has 34. There are three potential sources of noise in the dispersion measurement. The first results from random Teff errors, which can cause individual stars to be detrended improperly. The second source is uncertainty in the metallicity of the cluster. If I detrend data with a SSM calculated for an improper [Fe/H], the pre-MS mass trend will be improperly removed. Finally, random errors in EW(Li) measurements will impart scatter. To assess the impact of these factors, I randomly generated 1000 stars within each Teff bin. The stars were assigned true A(Li)s based on their temperature, by interpolating in a SSM

LDP calculated with solar metallicity. I assigned each an observed Teﬀ error by randomly sampling a Gaussian with σ = 50K, and propagated this error into the observed A(Li). When this was done, I assigned each a σ = 10% Gaussian random EW(Li) error, and altered the inferred A(Li) according. I believe 10% to

48 be a conservative estimate of the relative error of Li EWs within a single sample; although larger systematic errors between different studies may be present, this does not impact a dispersion measurement for a single data set. Finally, I detrended the stars against a SSM calculated with [Fe/H] = ±0.1 dex from the fiducial value. I performed this experiment 1000 times, and calculated the standard deviation within each bin. I found that in no case did the additional dispersion exceed 0.12 dex, implying that this method is reasonably stable to errors in photometry, spectroscopy, and composition. I represent this fiducial noise floor by the dashed lines in each right-hand panel of Fig. 2.8. Since the quality of observations vary between data sets, this calcuation is not accurate for all clusters. In particular, EW errors are more significant for Li-poor stars, which are more common in cool Teff bins and old clusters. I find that from the hot to cold bins, increasing the uncertainty in Teff to 100K increases the noise floor by 0.025, 0.003, 0.004, and 0.023 in the 6200K,

5900K, 5600K and 5300K bins respectively. An additional 0.1 dex uncertainty in

[Fe/H] increases the noise ﬂoor by 0.003, 0.013, 0.020, and 0.063 in the same bins.

An additional 10% uncertainty in EW(Li) increases the noise ﬂoor by 0.089, 0.104,

0.107, and 0.091, in the same bins.

This ﬁgure demonstrates that dispersion is a generic feature of cluster LDPs at all ages. It is present in some large samples as early as 100-200 Myr, suggesting an early origin for the dispersion in some clusters. However, dispersion is undetected in some very old clusters. This could be because of the small number of stars with

49 known Li abundances in some clusters, or could imply that dispersion is not present in all systems. Clusters of equal age can show significantly different dispersions, and some clusters of dissimilar age show equivalent dispersions; this again could be due to the quality of the data sets, or signify cosmic variance. To examine the mean trends, I grouped the clusters into 4 age bins, calculated the mean dispersion within each, weighing data points by their sample size, and plotted the results as purple squares. The 6200K bin shows that dispersion is, on average, present at all ages, but does not necessarily grow over time. Even the cluster with the greatest dispersion, M67, agrees with the mean trend in this Teff range. The 5900K bin shows the most clusters are confined to a small band of dispersions, with only a few outliers. There is again little evidence of dispersion evolution in this range, although some outlying clusters are now apparent. The 5600K bin shows a more convincing rising trend, with the five largest samples rising over time; this is evidence that in this temperature range, the spread can increase along the MS. Finally, in the coolest bin I see a trend of decreasing dispersion with age. This is related to the large dispersion at fixed temperature seen in young systems, such as the Pleiades, which is suppressed in some intermediate-age systems, such as the Hyades. This implies that the most abundant stars at ZAMS (i.e. the fastest rotating) deplete Li more rapidly on the MS, such that the dispersion decreases over the first Gyr of main sequence evolution.

50 While M67 hosts the clearest Li scatter of any cluster, both in Teﬀ space and in mass space (e.g. Pace et al. 2012), the precise magnitude of the M67 dispersion is uncertain due to the imprecision of EW measurements in low abundance stars.

The true dispersion at the solar temperature in M67 (ﬁnal point in the 5800K bin) may be smaller than the measured dispersion if the upper limits are unreasonably low (Onehag¨ et al. 2011, e.g.). Conversely, the true dispersion may be larger if the actual abundance of these stars is signiﬁcantly below the observed limit. Higher S/N

Li data is clearly required to deﬁnitely characterize the M67 distribution.

An obvious question is raised by these plots: if the magnitude of dispersion truly varies from cluster to cluster, what is the source of this variance? One possibility is that the dispersion is related to the initial conditions of open clusters.

The early onset of dispersion described above is qualitatively consistent with the picture expected from rotational-mixing. The greatest spread in rotation rates occurs in the ﬁrst 200 Myr of the MS, so the largest degree of diﬀerential depletion would occur then. If the initial AM distribution can vary between clusters, then the Li dispersion that ultimately develops will vary as well. Open cluster rotation distributions typically show a narrow, densely populated band of converged rotators, and a sparsely populated tail towards more rapid rotation (e.g. M37 − Hartman et al. 2009; Pleiades − Hartman et al. 2010; Irwin & Bouvier 2009, and refs. therein). In this paradigm, the tail would produce the most Li-depleted objects, and the converged stars should show smaller, nearly uniform depletion factors. If

51 rotation is truly responsible for the cosmic variance in LDPs, then the fractional size of the rapid-rotator tail must be what diﬀerentiates high-dispersion clusters from low-dispersion clusters. However, given the small fraction of stars in the high-velocity tail in some clusters (e.g. Hartman et al. 2010, ﬁg. 14), large Li data sets may be needed for a robust measure of the dispersion. Another cluster initial condition that could contribute to the scatter is the number density of members. Stellar mergers are more likely in dense environments, and low mass stars that undergo normal depletion on the pre-MS, then merge into a higher mass star during the MS, would appear scattered below the mean trend in a present day LDP. While both of these scenarios are qualitatively sensible, detailed work must be done to establish their predictions for the range of plausible resulting Li patterns.

Finally, does the dispersion of Li correlate with other observables? Another light element that can be destroyed by mixing on the main sequence is beryllium. One would expect a corresponding spread of this element, which burns at a temperature of 3.5 million K, to be present in clusters with a large Li dispersion. Since Be survives to deeper layers in stars, the expected pattern for a given theoretical scenario will diﬀer between the elements, but should correlate. Therefore, dual data sets would place additional constraints on theoretical models seeking to explain main sequence dispersion. Additionally, rich rotation data sets in clusters with extensive

Li measurements have recently been obtained (M34 − Meibom et al. 2011; Pleiades

− Hartman et al. 2010). If the early AM distributions of clusters impact their Li

52 evolution, this may be observable in the present day rotation patterns of clusters with diﬀering LDPs.

2.6. Empirical Test of the Metallicity Dependence of Lithium

Depletion

2.6.1. Does Metallicity Impact Pre-MS Lithium Depletion?

Depletion in SSMs is a strong function of metallicity, since the surface CZ deepens on the pre-MS with increasing opacity. I have relied on the validity of this strong theoretical prediction in my calculation of the lithium anomaly. However, the existence of this metallicity eﬀect has been questioned by some authors from the observational side. Jeﬀries & James (1999) argued that the Li distribution of the open cluster Blanco 1 is identical to that of the similarly aged Pleiades, despite claims that it is 0.1-0.2 dex richer in metals. Sestito et al. (2003) reached a similar conclusion from a comparison of NGC 6475 and M 34. While puzzling at the time, these results have since been called into question by more recent calculations of the iron abundance of Blanco 1 (+0.04 ± 0.02; Ford et al. 2005), NGC 6475 (+0.03 ±

0.02; Villanova et al. 2009), and M 34 (+0.07 ± 0.04; Schuler et al. 2003).

Nevertheless, there remain open clusters of diﬀering [Fe/H] with apparently similar empirical LDPs, particularly the trio of ∼ 1.5 Gyr old clusters: NGC

752, NGC 3680, and IC 4651 (Sestito et al. 2004; Anthony-Twarog et al. 2009).

53 Diﬀerential depletion imprinted at ZAMS should persist as cluster evolve on the MS, so these authors argue that the similarity of the current Li patterns suggests pre-MS depletion cannot depend on [Fe/H]. Can this result be reconciled with standard stellar theory? In this section, I address this question by comparing these cluster patterns in absolute and anomaly space. I apply this same analysis to the Hyades and NGC 6633, two 600 Myr old clusters with substantial composition diﬀerences. If metallicity is an important factor in determining ZAMS abundance, the metal poor cluster patterns should lie above the metal rich patterns in absolute space, and on top of them in anomaly space.

I ﬁrst examine the LDPs of the three clusters mentioned above: NGC 752,

NGC 3680, and IC 4651. Iron abundances and ages for each are reported by

Anthony-Twarog et al. (2009): [Fe/H] = −0.05 and age = 1.45 Gyr for NGC 752,

[Fe/H] = −0.08 and age = 1.75 Gyr for NGC 3680, and [Fe/H] = +0.13 and age

= 1.5 Gyr for IC 4651. These abundances are due to high resolution spectroscopy, and appear robust in their relative values. I draw BV photometry and Li EWs from

Sestito et al. (2004) for NGC 752, from Anthony-Twarog et al. (2009) for NGC 3680, and from Randich et al. (2000) for IC 4651. I synthesize this data into eﬀective temperatures and A(Li)s using the method described in Section 2.2.2, and compute

SSM Li predictions using the methodology of Section 2.2.3.

These data are presented in the left panel of Fig. 2.9. NGC 3680 (ﬁlled black) and NGC 752 (empty black) have similar Fe abundances, and unsurprisingly show

54 similar median Li trends. IC 4651 (empty red) is ∼ 0.2 dex richer in Fe, but lines up well in absolute space with the other clusters. At face value this appears surprising, given the assumption of a strong [Fe/H] dependence of ZAMS Li patterns.

However, since IC 4651 data only exists for stars with Teﬀ ∼> 5800K, a regime that is particularly insensitive to composition (∆[Fe/H] = 0.2 dex → ∆A(Li) = 0.3 dex at 5800K; Section 2.3.1), modest scatter may obscure the relative depletion signal. The bottom left shows the results of Kolmogorov-Smirnov (K-S) tests, which demonstrates that in each case, the clusters cannot be statistically distinguished.

This supports the visual impression that the cluster LDPs are similar.

The right hand side shows the same data detrended with respect to each clusters’ SSM prediction, as described in Section 2.4. In the anomaly plane, the relative cluster distributions remain similar. NGC 3680 and NGC 752 have shifted very little with respect to one another, but IC 4651 has trended upward relative to the other clusters. The locus of IC 4651 data appears to lie near the top of the distribution of NGC 3680 and NGC 752, but this visual impression is largely due to a single data point at the top right of the IC 4651 distribution. These clusters still cannot be distinguished by a K-S test, and show no statistical improvement over the comparison in absolute space. This suggests that the cluster LDPs appear similar because they are in a temperature range that is particularly insensitive to metallicity. Furthermore, Cummings (2011) found that more Fe rich clusters were born with a higher initial Li abundance, so IC 4651 may have begun life with

55 slightly more Li, and suﬀered slightly higher depletion to wind up at a similar location in anomaly space at 1.5 Gyr. Given the weak dependence of the models on metallicity in this regime, the similarity of the LDPs in anomaly space, and the remaining uncertainties in initial Li abundance and relative [Fe/H], I conclude that these clusters do not convincingly demonstrate that composition is unimportant in determining the ZAMS Li pattern.

Another pair of clusters that are similar in age, and whose relative metallicities have been well-determined, are the Hyades and NGC 6633. The latter is a ∼ 600

Myr old (Strobel 1991) open cluster with an iron abundance that is 0.206 ± 0.40 dex less than the Hyades (Jeﬀries et al. 2002). This is a secure relative value, since both used high resolution spectroscopy, and the analysis of NGC 6633 was carried out in precisely the same way as the analysis that derived the benchmark Hyades data. A substantial Li data set is available for this cluster, due as well to (Jeﬀries et al. 2002).

With my prior Hyades [Fe/H] of +0.135, I derive [Fe/H] = −0.071 ± 0.040 for NGC

6633. I apply a similar analysis as for the above cluster trio, and present the results in Fig. 2.10. The left panel shows the Hyades (empty circles) and NGC 6633 (ﬁlled circles) abundances compared with one another. The Li distributions are similar above 5700K, just as in the above case, but quite dissimilar below 5300K. A K-S test shows that the probability that these data sets are equivalent is < 0.5%. Next,

I detrend the clusters and plot them in the right panel of Fig. 2.10. In this plane, both the warm and cool NGC 6633 stars appear to lie on top of the Hyades stars. A

56 K-S test reveals that the populations can no longer be statistically distinguished (∼

0.72 rejection probability), suggesting that the magnitude of diﬀerential depletion between these two clusters is accurately predicted by SSMs. This strongly implicates metallicity as the source of the diﬀerence between the Hyades and NGC 6633 LDPs.

The similarity of these two clusters in detrended space, and the paucity of secure counter-examples, leads us to conclude that metallicity does impact pre-MS

Li destruction. This implies that my calculation of the lithium anomaly is a valid measure of main sequence depletion. To solidify this result, and to quantify the magnitude of the pre-MS metallicity effect, the LDPs of additional groups of similar-aged clusters must be compared. Future work should focus on obtaining large samples of Li measurements of cooler stars. The temperature range is critical, since SSMs predict that a 0.2 dex difference in iron abundance leads to a 0.2 dex difference in Li at 6000K, but a 1.2 dex difference at 5000K. The former may not be detectable, but the latter certainly is. Furthermore, accurate relative compositions must be determined if one hopes to compare the LDPs of equal-age clusters. Heiter et al. (2013) demonstrates that differences in the analysis method are a larger source of error in [Fe/H] determinations than differences in the quality of observations when using high-resolution spectra. This suggests that a uniform re-analysis of high-resolution Fe observations may help to resolve this issue.

57 2.6.2. Lithium as a Tracer of Cluster Metallicity

The extreme sensitivity of Li depletion to metallicity opens the exciting possibility of using Li to infer cluster composition. If the detrended LDPs of equal age clusters are morphologically similar, then the composition of a cluster can be constrained by ﬁnding the SSM LDP that produces the best agreement with a cluster of known composition in anomaly space. I tested this concept by estimating the metallicity of the ∼ 700 Myr old Praesepe (Salaris et al. 2004) using the well-constrained, and similar aged, Hyades. The [Fe/H] of Praesepe has been a controversial subject: it has been measured as low as +0.04 ± 0.04 (Boesgaard & Friel 1990), and as high as

+0.27 ± 0.10 (Pace et al. 2008), with the most recent, high-resolution spectroscopic study ﬁnding +0.12 ± 0.04 (Boesgaard et al. 2013). I ﬁrst analyzed Praesepe Li EW and BV data from (Soderblom et al. 1993b) using the methods described in Section

2.2.2, with the metallicity of (Boesgaard et al. 2013). Using the machinery of Section

2.2.3, I created 0.7 Gyr SSM predictions for [Fe/H]s ranging from -0.30 to 0.30, in steps of 0.01, and detrended the Praesepe data with each. I then compared each detrended pattern with the Hyades anomaly pattern. To establish a quantitative measure of agreement, I calculated the difference in the median anomalies of the two clusters in three Teff bins: 5500-5700K, 5700-5900K, and 5900-6100K. I then weighted each point by the MAD of the data around the median, and added the differences in quadrature. The resulting [Fe/H] vs. goodness-of-fit curve is shown in Fig. 2.11, with lower goodness-of-fit values demonstrating superior agreement.

58 As can be seen, my best ﬁt [Fe/H] = +0.16, consistent at 1σ with the most recent high-resolution spectroscopy estimate for Praesepe.

I note a few caveats to this method. First, there is a partial degeneracy between cluster [Fe/H] and the initial Li abundance of its members. I assessed the impact of this effect by changing the initial Li abundance of my Praesepe models, and recalculating the best fit [Fe/H]. I found that the best fit [Fe/H] changed to +0.10 when the initial Praesepe abundance was decreased by 0.1 dex, and +0.20 when the initial Praesepe abundance was increased by 0.1 dex. I expect that this is a minor effect, since clusters must be coeval for this method to work. Furthermore, an uncertainty of ± 0.05 dex in [Fe/H], from an uncertainty of ± 0.1 dex in initial Li abundance, is small compared to quoted errors on most spectroscopic metallicities.

Second, I did not re-derive the stellar parameters of Praesepe members for each

[Fe/H]. The Teff scale is somewhat sensitive to [Fe/H], so a more careful calculation should take this effect into account when determining goodness of fit. Nevertheless, this proof-of-concept demonstrates that this method merits further attention.

In summary, I have shown that the metal abundance of stars is a crucial factor in shaping pre-MS Li abundance patterns. A trio of 2 Gyr old clusters with different compositions but similar LDPs were shown to be consistent with the prediction of additional depletion in metal-rich stars. Since the stars in these samples are all in an Teff regime where the metallicity effect is weak, their apparent similarity in A(Li) space does not constitute evidence against the standard picture. On the other hand,

59 two 600 Myr old clusters of differing composition were shown to have significantly different average Li abundances in absolute space. Once detrended into anomaly space, the clusters became statistically indistinguishable, demonstrating that SSMs accurately predicted the difference between their patterns. Finally, I have shown that the strong dependence of LDPs on composition can be used to constrain the metallicity of open clusters, by comparing them to an equal-age clusters with well known metallicity.

2.7. Summary

I have attempted to reconcile SSMs with observations at the ZAMS, and use their predictions to uncover the behavior of Li on the MS. Through my investigation, I have reached several conclusions about SSM Li predictions, the progression of Li destruction on the MS, and the development of dispersion in cluster Li patterns.

1) Pre-MS Li depletion in solar analogs is accurately predicted by standard stellar models. SSM Li depletion on the pre-MS is highly sensitive to the assumed input physics. Adopting alternate choices of physical inputs, such as the cluster metal content, the equation of state tables, the opacity tables, and the 7Li(p, α)α interaction cross-section, induces substantial changes in the predicted Li pattern at the ZAMS. I measured the impact of each physical input on pre-MS Li destruction, and derived an error band. Within this band, the distribution of solar analogs in the near-ZAMS Pleiades agrees with standard predictions. This allows us to perform an

60 ad hoc calibration of the physics in my models, by forcing them to accurately predict the Li pattern of the Pleiades. With these calibrated models, I can predict the

ZAMS distribution of other clusters by applying the metallicity scalings predicted by standard theory. This allows us to detrend the metallicity-dependent pre-MS depletion out of empirical cluster data, leaving behind the depletion occurring on the MS, which is insensitive to metallicity and input physics.

2) Dispersion is a general feature of Li patterns on the MS. I detrend the strong pre-MS mass trends out of the full suite of clusters considered by Sestito & Randich

(2005), and infer the dispersion within Teff bins centered at 6200K, 5900K, 5600K, and 5300K. Within each bin, a dispersion is found to emerge in some clusters during the first 200 Myr. However, the evolution of this dispersion over time depends on the Teff of the bin. The dispersion present at 100 Myr shows little evolution for the

6200K bin, but a marginal rising trend is seen in the 5900K bin, a more convincing rising trend is seen in the 5600K bin, and a decreasing trend is seen in the 5300K bin. The degree of dispersion may diﬀer between clusters of equal age. If true, this points to true cosmic variance in the initial conditions of open clusters. Finally,

I show that the depletion mass trends identiﬁed by Sestito & Randich (2005) are stronger in anomaly space than in absolute space.

3) The rate of main sequence Li depletion is a strong function of mass. I isolated the main sequence depletion signal from two well-studied main sequence clusters, the Hyades and M67, by subtracting the predicted ZAMS depletion factors from the

61 empirical data of the clusters. Then, I measured the average rate of main sequence depletion over their lifetimes by comparing the median abundances of the clusters at several temperatures. This is the first measure of main sequence Li depletion that is not biased by [Fe/H]-dependent, differential pre-MS depletion. I find that low mass stars deplete Li significantly faster than high mass stars on the MS, placing strong constraints on propose mechanisms of main sequence mixing. I also find that the rate of main sequence depletion decays over time, in agreement with previous work. These are both robust conclusions, since they persist even in the presence of significant differences in the initial Li abundance of my benchmark clusters.

4) Cluster metallicity is crucial in shaping ZAMS Li abundance patterns.

Some authors have argued that empirical studies of open clusters rule out a strong dependence of pre-MS Li depletion on metallicity. I show that an outstanding case for this argument, involving three 2 Gyr old clusters with dissimilar composition but similar Li patterns, does not rule out the theoretically predicted pre-MS metallicity dependence. I then analyze two ∼ 600 Myr old clusters that diﬀer in [Fe/H] by 0.2 dex, and show substantial diﬀerences in their cool star Li patterns. Their absolute

Li distributions are shown to be different at high confidence. When the metallicity dependence of SSMs is corrected for, they become statistically indistinguishable. My models accurately predicted the difference in abundances of these clusters, strongly implicating metallicity as an important factor. Since cluster LDPs are sensitive to metallicity, sizable Li data sets can be used to estimate the metallicity of their host

62 clusters by comparison with a similar aged cluster of well known composition. I apply this method to the Praesepe cluster, and ﬁnd a best-ﬁt [Fe/H] = +0.16, in 1σ agreement with the recent high-resolution spectroscopic measurement of +0.12 ±

0.04 (Boesgaard et al. 2013).

63 0.25 0.25 4.0

Teff Mapping Curves of Growth 0.20 Systematics Systematics 0.20 3.5 3.0 A(Li) 07 ′ 0.15 0.15 An 2.5 S 93

0.10 0.10 − A(Li) 2.0 A(Li) −

A(Li) 10

C 0.05 0.05 1.5 S 03

A(Li) 0.00 0.00 1.0

0.5 −0.05 −0.05 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 6500 6000 5500 5000 4500 A(Li)C10 A(Li)S93 Effective Temperature (K)

Fig. 2.1.— Systematic errors in lithium and temperature derivations. Graphic illustrations of the potential systematic errors in my Li abundance analysis (see Section 2.2.2 for details). Left: Hyades abundances derived with the Casagrande et al. (2010) Teff scale versus the offset between that sample and Hyades abundances derived with the An et al. (2007) Teff scale. Both use S93 curves of growth. Center: Hyades abundances derived with the Soderblom et al. (1993a) curves of growth versus the offset between that sample and Hyades abundances derived with the Steinhauer (2003) curves of growth. Both use Casagrande et al. (2010) Teff scale. Right: The effect of Teff errors on derived abundances. The line shows the direction and magnitude of data movement resulting from ±100K errors in the data. The correlation between Teff and A(Li) causes the points to move diagonally, largely preserving the LDP shape in the cool star regime.

64 Effective Temperature (K) Effective Temperature (K) 6500 6000 5500 5000 6500 6000 5500 5000

Equation of State Model Atmosphere 1.5 Hi-Temp Opacity 1.5 Lo-Temp Opacity Li7 Cross Section ±∆Y/∆Z

1.0 Asplund 09 (Z/X) ⊙ 1.0 Fiducial Model ∆ A(Li)

A(Li) 0.5 0.5 ∆

0.0 0.0

Individual errors from Individuals error from 0 to 0.12 Gyr 0.12 to 2 Gyr −0.5 −0.5

3.0 0.147 Upper Error 0.004 3.0 0.255 0.003 Bound 0.007 Upper Error 2.5 0.418 2.5 0.021 Bound 0.638 2.0 2.0 Lower Error 0.950 Lower Error 0.095 1.5 Bound Bound 1.5 A(Li) A(Li) 1.0 1.409 0.256 1.0

0.5 0.5

0.0 0.0 Cumulative error from 2.123 Cumulative error from 0 to 0.12 Gyr 0.12 to 2 Gyr 0.483 −0.5 −0.5 6500 6000 5500 5000 6500 6000 5500 5000 Effective Temperature (K) Effective Temperature (K)

Fig. 2.2.— Dependence of theoretical lithium depletion on model physics. The effect of adopting alternate model physics on SSM Li depletion during the pre-MS (left) and the main sequence (right). Top: The difference between the fiducial Li pattern and the Li pattern resulting from exchanging one physical input for another commonly adopted in the literature. Colors correspond to different physical inputs, as enumerated in the key. The errors in the left panel develop during the first 0.12 Gyr, and those on the right develop between 0.12 and 2 Gyr. Depletion on the pre- MS is highly sensitive to errors in the physical inputs, but main sequence depletion is stable. Bottom: The quadrature sum of the errors as a function of Teff . Any Li curve lying in the shaded region is a statistically acceptable Li depletion pattern.

65 Systematic Fiducial Model Systematic Lower Bound Upper Bound 1 1

0 0 ∆ A(Li) A(Li)

−∆ 1 −1

−2 −2

6400 6000 5600 5200 6400 6000 5600 5200 6400 6000 5600 5200 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 2.3.— Influence of systematics on detrended cluster lithium. The impact of uncertainties in my adopted model physics on the relative locations of clusters in anomaly space. Lithium data for the Pleiades (red circles), Hyades (blue triangles), and M67 (purple squares) are transformed into anomaly space using SSM Li predictions from the bottom envelope of the systematic error band (left), the fiducial model (center), and the top envelope of the systematic error band (right), as illustrated by Fig. 2.2. The absolute difference between cluster anomaly patterns is sensitive to the adopted model physics, so the models must be calibrated using empirical data. Arrows denote upper limits.

66 3.4

3.2

3.0

2.8

A(Li) 2.6

2.4

2.2

2.0 6200 6000 5800 5600 5400 Effective Temperature (K)

Fig. 2.4.— Fitting the Pleiades lithium pattern. Pleiades systematic error band from Fig. 2.2 over-plotted with Pleiades data in the range 5500K-6100K. Although any LDP lying in the shaded region is statistically acceptable given the uncertainties in my model physics, the dashed blue line represents the best ﬁt to the data. This serves us a calibration of the input physics in my models.

67 3.5 3.5 Pleiades Hyades M67 3.0 125 Myr 625 Myr 3.9 Gyr 3.0

2.5 2.5

2.0 2.0

A(Li) 1.5 1.5

1.0 1.0

0.5 0.5

0.0 0.0 6500 6000 5500 5000 4500 6500 6000 5500 5000 4500 6500 6000 5500 5000 4500 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 2.5.— Corrected standard model lithium depletion for benchmark clusters. Solid black lines represent SSM Li patterns, whose input physics has been calibrated on the Pleiades (Section 2.4). Dashed black Li patterns represent the ﬁducial LDPs from Fig. 1.2. The vertical dashed lines denote the Teﬀ regime employed to calibrate my model physics. Red, blue and purple data points are the same as in Fig. 1.2.

68 1.0

0.5

0.0

−0.5

−1.0 Lithium Anomaly Lithium

−1.5

−2.0

−2.5 6300-6100K 6100-5900K 5900-5700K 5700-5500K 5350-5150K 6200 6000 5800 5600 5400 5200 Effective Temperature (K)

Fig. 2.6.— Final lithium anomaly measurements for benchmark clusters. The ﬁnal Li anomaly measurements for my benchmark clusters. The Pleiades (red circles), Hyades (blue triangles) and M67 (purple squares) data have been detrended with respect to their calibrated SSM LDPs, described in Section 2.4. Light red, light blue, and light purple stars are the average Teﬀ and median lithium anomaly in each bin (denoted by black dashed lines) for the Pleiades, Hyades and M67, respectively. An inverted triangle denotes an upper limit to the median anomaly. The solar lithium abundance is also shown, and arrows denote upper limits.

69 6400 6200 6000 5800 5600 5400

0.0 Sestito & Randich (2005) Somers & Pinsonneault (2014)

0.5 A(Li) ∆

1.0 Pleiades → Hyades

0.0

0.5

1.0 A(Li)

∆ 1.5

2.0

Pleiades → M67 2.5 6400 6200 6000 5800 5600 5400 Effective Temperature (K)

Fig. 2.7.— Comparison with literature lithium depletion measurements. A comparison of the lithium anomaly measurements of this work with the measurements of Sestito & Randich (2005). Red squares represent this work’s calculation of the lithium anomaly for the Hyades (top) and M67 (bottom). Blue circles represent the diﬀerence between the Sestito & Randich (2005) bin containing the Pleiades and the bins containing the Hyades (top) and M67 (bottom). Sestito & Randich (2005) reported the average of the upper and lower envelopes for M67, so I include both sets of points, and show the region interior. Triangles represent upper limits.

70 Age (Myr) Age (Myr) 101 102 103 104 101 102 103 104

6200 ±150K 0.6 1 3 0.5 0.4

0 2 A(Li) 0.3 −1 1 0.2 Dispersion Li Anomaly Li |dA(Li)/dlog(Age)| = −0.211 0.1 −2 0 |dA(Li)/dlog(Age)| = −0.225 0.0 6200 ±150K

5900 ±150K 0.6 1 3 0.5 0.4

0 2 A(Li) 0.3 −1 1 0.2 Dispersion Li Anomaly Li |dA(Li)/dlog(Age)| = −0.398 0.1 −2 0 |dA(Li)/dlog(Age)| = −0.557 0.0 5900 ±150K

Age (Myr) 5600 ±100K 0.6 Age (Myr) 1 3 0.5 0.4

0 2 A(Li) 0.3 −1 1 0.2 Dispersion Li Anomaly Li |dA(Li)/dlog(Age)| = −0.612 0.1 −2 0 |dA(Li)/dlog(Age)| = −0.883 0.0 5600 ±100K

Age (Myr) 5300 ±100K 0.6 Age (Myr) 1 0.5 0 0.4 0.3 −1 0.2 Dispersion Li Anomaly Li 0.1 −2 |dA(Li)/dlog(Age)| = −1.251 0.0 5300 ±100K 101 102 103 104 101 102 103 104 Age (Myr) Age (Myr)

Fig. 2.8.— Timescales of lithium depletion and dispersion. The timescales of Li depletion in anomaly space, and the development of Li dispersion at fixed mass. Left: Li data from Sestito & Randich (2005) (red points, right scale) compared to the same data detrended with the methods of Section 2.4 (black points). The red and black lines show the best fit power laws calculated up to 2 Gyr, with slopes reported in the lower left of each panel. The detrended analysis reveals a stronger mass trend than the absolute analysis. Right: The dispersion of each cluster considered in the left, with the sample size represented by the radius of the individual points. The dashed lines represent the absolute noise floor, and purple squares represent a weighted average of points within each age bin. Dispersion is found to emerge early in many clusters, and stay relatively constant for late F stars, rise marginally over time in solar analogs, and decrease up to 1 Gyr in cool stars.

71 3.5 0.5 NGC3680 NGC752 3.0 IC4651 0.0

2.5 −0.5 Lithium Anomaly

2.0 −1.0

A(Li) 1.5 −1.5

1.0 −2.0

3680/752 Reject % = 0.566691 3680/752 Reject % = 0.566691 0.5 −2.5 3680/4651 Reject % = 0.600105 3680/4651 Reject % = 0.412897 752/4651 Reject % = 0.848154 752/4651 Reject % = 0.982760 0.0 −3.0 6500 6100 5700 5300 4900 6500 6100 5700 5300 4900 Effective Temperature (K) Effective Temperature (K)

Fig. 2.9.— Impact of metallicity on lithium data at 2 Gyr. A comparison of three 2 Gyr old clusters in A(Li) and Li anomaly space, which demonstrates that they do not rule out metallicity eﬀects on the pre-MS. NGC 3680 (ﬁlled black circles), NGC 752 (empty squares), and IC 4651 (empty red circles) data and shown alongside their SSM predictions (solid black, dashed black, and dashed red lines, respectively) in the left panel, and subtracted from their SSM predictions in the right. IC 4651 is ∼ 0.2 dex richer in metals, but lines up well with the other clusters in both absolute and anomaly space. The inconclusive rejection probabilities shown in the bottom left of each panel imply that in neither case can the abundance distributions be statistically distinguished.

72 3.5 0.5 Hyades NGC6633 3.0 0.0

2.5 −0.5 Lithium Anomaly

2.0 −1.0

A(Li) 1.5 −1.5

1.0 −2.0

0.5 −2.5 KS Reject % = 0.004871 KS Reject % = 0.720980 0.0 −3.0 6500 6100 5700 5300 4900 6500 6100 5700 5300 4900 Effective Temperature (K) Effective Temperature (K)

Fig. 2.10.— Impact of metallicity on lithium data at 600 Myr. A comparison of two 600 Myr old clusters in A(Li) and anomaly space, which demonstrates that metallicity likely impacted their pre-MS abundance distributions. Hyades (empty) and NGC 6633 (ﬁlled) data are shown alongside their SSM predictions (dashed black and solid black lines, respectively) in the left, and subtracted from their SSM predictions in the right. The null hypothesis that the Li distributions are equal can be rejected at high conﬁdence in A(Li) space, but cannot be rejected in detrended space. This suggests that SSMs accurately predict the relative depletion of these clusters.

73 5 2 ) K 6000

+( 4 2 ) K 5800

+( 3 2 ) K 5600 (

1 Goodness of Fit = = Fit of Goodness −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 [Fe/H]

Fig. 2.11.— Lithium as a metallicity tracer in old clusters. The quality of agreement between detrended Hyades Li data and detrended Praesepe Li data, versus the metallicity assumed when detrending Praesepe. The best ﬁt [Fe/H] is at +0.16, implying that this is the most likely metallicity for the Praesepe, according to the method described in Section 2.6.2. This is in good agreement with the most recent high-resolution spectroscopic measurement of +0.12 ± 0.04 (Boesgaard et al. 2013).

74 Error Source Fiducial Alternate Alternate ∆A(Li) at... Choice1 Choice Ref.2 6500K 6000K 5500K 5000K 4500K

Equation of State OPAL 2006 SCV (1) +0.03 +0.11 +0.31 +0.64 +0.83 Model Atmosphere Kurucz Allard (2) −0.01 −0.02 +0.01 +0.16 +0.71 High-Temp Opacity OP17 OPAL17 (3) +0.01 +0.02 +0.03 −0.04 −0.40 Low-Temp Opacity Alex 2006 Alex 1995 (4) −0.03 −0.08 −0.17 −0.32 −0.67 7Li(p, α)α Cross Section Lamia 2012 ± 9% ±0.01 ±0.03 ±0.08 ±0.18 ±0.49 Initial Solar Composition GS98 Asp09 (5) +0.04 +0.14 +0.37 +0.90 +2.01

75 ∆Y/∆Z 1.13 ± 0.2 ±0.00 ±+0.01 ±0.03 ±0.09 ±0.28

Total + Error +0.05 +0.18 +0.49 +1.13 +2.35 Total − Error −0.03 −0.09 −0.19 −0.38 −0.97

1Fiducial citations may be found in Section 2.2.3 2Alternate citations are as follow: (1) Saumon et al. 1995; (2) Allard et al. 1997; (3) Iglesias & Rogers 1996; (4) Alexander & Ferguson 1994; (5) Asplund et al. 2009

Table 2.1. Pre-MS Systematic Errors Curves of Cluster Anomaly At...3 Growth 6200K 6000K 5800K 5600K 5250K

Soderblom Pleiades −0.157 ± 0.039 −0.012 ± 0.052 0.033 ± 0.042 0.021 ± 0.047 0.291 ± 0.177 Hyades −0.177 ± 0.119 −0.195 ± 0.030 −0.320 ± 0.041 −0.964 ± 0.103 < −1.345 M67 −0.627 ± 0.089 −0.913 ± 0.075 −1.697 ± 0.134 < −2.415 −

Pleiades → Hyades 0.020 ± 0.125 0.183 ± 0.060 0.353 ± 0.058 0.984 ± 0.113 > 1.636 Pleiades → M67 0.471 ± 0.097 0.901 ± 0.092 1.730 ± 0.141 > 2.436 − Hyades → M67 0.451 ± 0.149 0.718 ± 0.081 1.377 ± 0.140 > 1.451 − 76 Steinhauer Pleaides −0.141 ± 0.038 0.001 ± 0.052 0.039 ± 0.041 0.018 ± 0.046 0.259 ± 0.174 Hyades −0.162 ± 0.127 −0.177 ± 0.031 −0.306 ± 0.042 −1.085 ± 0.124 < −1.549 M67 −0.623 ± 0.095 −0.950 ± 0.083 −1.891 ± 0.127 < −1.980 −

Pleaides → Hyades 0.021 ± 0.132 0.177 ± 0.060 0.345 ± 0.059 1.103 ± 0.133 > 1.808 Pleaides → M67 0.482 ± 0.103 0.950 ± 0.098 1.930 ± 0.133 > 1.998 − Hyades → M67 0.461 ± 0.159 0.773 ± 0.089 1.585 ± 0.133 > 0.895 −

3Absolute and relative main sequence Li depletion factors. The upper values were calculated with the curves of growth of Soderblom et al. (1993a), and the lower values with the curves of growth of Steinhauer (2003).

Table 2.2. Benchmark Cluster Lithium Anomalies Chapter 3: Suppressed Convection and the Radius Anomaly as the Origin of the Lithium Dispersion in Young Cool Dwarfs?

3.1. Background

An intriguing feature of the Pleiades Li pattern that standard stellar theory cannot explain is the large abundance dispersion in stars cooler than 5500K. While SSMs predict that Li content is uniquely determined by age, mass, and composition, a spread of ∼1.5 dex in A(Li) was discovered in the cool Pleiads by Duncan & Jones

(1983), and later conﬁrmed by Soderblom et al. (1993a). The latter authors further showed that in this temperature regime, fast rotating stars are on average less depleted than their slow rotating counterparts. This is counter-intuitive, because rapid rotation is expected to, if anything, drive additional mixing, and therefore deplete more Li (Pinsonneault et al. 1989).

In the years since this discovery, several authors have attempted to explain this dispersion through surface effects that impact λ6708 Li I EWs. One possibility is chromospheric activity affecting the physics of line formation (Soderblom et al. 1993a; Carlsson et al. 1994; Stuik et al. 1997; Jeffries 1999). These authors argued that such an effect would impact other lines as well, such as the λ7699 K I

77 resonance line. Soderblom et al. (1993a) initially found a commensurate spread in the λ7699 K I and λ6707 Li I features in cool Pleiads, supporting the notion that the Li spread is spurious. However, high resolution spectra obtained by King et al. (2010) found no such scatter in the K I line, while conﬁrming a spread of ∼> a factor of 2 in the Li I line. Another potential eﬀect is rotational broadening of the

Li absorption line in rapid rotators (Margheim et al. 2002), which could naturally explain why fast spinning stars appear more abundant. However, Margheim (2007) later demonstrated that this cannot account for the full dispersion, since abundances derived though EW analysis and abundances derived through rotationally-broadened spectral synthesis agreed for the majority of the stars. Finally, if cool Pleiads have a large spot filling factor, the neutral Li I line could be enhanced, as the ionization fraction would be lower on large swaths of the surface. However, King et al. (2010) demonstrated that this cannot account for the full effect, as the maximum filling factors necessary to develop the observed EW spread would result in a V-band magnitude dispersion of ∼ 1 mag, a feature not observed in Pleiades HR diagrams.

It appears that a combination of these eﬀects may contribute to the total dispersion, but that a true underlying Li spread must exist.

Several mechanisms have been proposed as the cause of this eﬀect, as discussed in Section 1.3. In this chaper, I investigate a version of one such possibility: increased

PMS stellar radii impacting the internal temperature of young objects. The rate of pre-MS Li depletion in a star is exquisitely sensitive to TBCZ during the proto-stellar

78 epoch (Section 2.1). If some mechanism reduced TBCZ below its standard predictions during this time period, the Li depletion rate would be severely suppressed, and the ZAMS abundance would be higher than anticipated. Such inﬂated radii, of order 5-15%, have been observed in detached eclipsing binaries (DEBs; Popper 1997;

Torres & Ribas 2002; Ribas 2003; L´opez-Morales & Ribas 2005; L´opez-Morales

2007; Torres, Anderson, & Gim´enez 2010; Kraus et al. 2011; Irwin et al. 2011;

Feiden & Chaboyer 2012a; Stassun et al. 2012), are present in interferometric radius measurements of single field stars (Berger et al. 2006; Boyajian et al. 2008, 2012) and and have indeed been claimed in the Pleiades itself (Jackson & Jeffries 2014a). This effect has also been reported in solar analogs (e.g. Clausen et al. 2009) and pre-MS stars (Stassun et al. 2006; Stassun, Mathieu & Valenti 2007). The presence of such radius anomalies during the PMS would reduce the pressure in the central regions, thus decreasing the temperature required to maintain equilibrium, and slowing the rate of Li burning.

In Section 3.2, I brieﬂy discuss potential underlying causes of the radius anomaly, and describe how such anomalies could induce the observed Li pattern. In

Section 3.3, I present inﬂated stellar models of the Pleiades, and conclude that radius anomalies of the observed magnitude can suppress Li depletion by the required amount. In Section 3.4, I extend this analysis to six additional young clusters, and show that radius anomalies can explain the general pattern of LDP evolution on the pre-MS. In Section 3.5, I discuss implications of radius anomalies for pre-MS

79 evolutionary tracks, stellar initial mass function (IMF) measurements, and the ages of young open clusters. Finally, in Section 3.6 I summarize the results of this section and suggest directions for future study.

3.2. The Radius Anomaly

Radius anomalies have been observed in a large number of systems (see Section 2.1).

While the underlying cause is not yet understood, several explanations have been put forward. Accretion from a circumstellar disk may increase (Palla & Stahler 1992) or decrease (Baraﬀe et al. 2009) proto-stellar radii, depending on the assumptions about the system. While this is likely important on the pre-MS, the existence of radius anomalies at a few 100 Myr (e.g. YY Gem Torres & Ribas 2002), long after the T

Tauri phase, makes this unlikely to be the sole culprit. Theoretical expectations for a given system can also be greatly impacted by errors in metallicity, as stellar radii are sensitive to opacity. This is particularly apparent in interferometric measurements of M dwarfs (Berger et al. 2006), and may indicate missing opacity sources in very cool stars. Nevertheless, radius anomalies persist in many systems with very well measured composition (e.g. CM Dra Terrien et al. 2012), so this cannot fully explain the phenomenon.

A third possible explanation is the presence of magnetic ﬁelds in low-mass stars (Mullan & MacDonald 2001; Chabrier, Gallardo & Baraﬀe 2007; Morales et al. 2008; MacDonald & Mullan 2012; Feiden & Chaboyer 2013), which can impact

80 the radius in two ways. First, strong magnetic activity can increase the coverage of spots on the stellar surface, which reduces Teﬀ and puﬀs out the envelope (Andronov

& Pinsonneault 2004). Second, it can inhibit the efficiency of thermal convection, which creates a stronger radiative energy gradient to compensate for the reduced convective energy flux. This enhances the temperature gradient, which leads to a lower surface temperature and causes the radius to expand. This theory is supported by the discovery of a correlation between the radius anomaly and the strength of surface magnetic field proxies. Stars with stronger coronal activity, inferred from the ratio of X-ray (or Hα) to bolometric luminosity, show a larger fractional disagreement between their measured radii and SSM predictions (López-Morales

2007; Clausen et al. 2009; Stassun et al. 2012). Coronal activity also correlates with rotation on the main sequence (e.g. Wilson 1966; Kraft 1967; Fleming et al. 1989;

Bouvier 1990), implying that stars with the most inﬂated radii may also be the most rapidly rotating. Young clusters, such as the Orion Nebula Cluster, possess a large range of X-ray luminosities (Preibisch et al. 2005) and rotation rates (Stassun et al. 1999; Herbst et al. 2001, 2002) at ﬁxed mass, suggesting that a range of radius anomalies, as large as 0-15%, may be present in young clusters.

This presents a plausible explanation for the Li spread in the Pleiades: the most rapidly rotating stars are puﬀed up relative to standard models on the pre-MS, causing their central temperatures to decrease. This greatly inhibits Li depletion during the pre-MS burning phase, and so they lie signiﬁcantly above the SSM LDP

81 at the age of the Pleiades. The slowest rotating stars have no radius anomaly, burn

Li at a rate consistent with standard predictions, and therefore lie close to the SSM

LDP. The result is a range of abundances between the standard prediction for a given

Teﬀ , and the Li abundance produced by a star with the maximal radius anomaly.

Moreover, the most inﬂated stars are also the most rapidly rotating, creating an observable correlation between Li abundance and surface rotation rate at 120 Myr.

This is an elegant solution to this problem, as an empirically observed radius eﬀect may simultaneously explain the abundance spread, why the median lies above the standard prediction, and why fast rotating stars are less depleted.

3.3. Lithium and Radius in the Pleiades

Before this possibility can be considered further, I must first investigate whether radius anomalies of the observed magnitude can suppress Li burning by the required amount. To do this, I calculate standard stellar models with inflated radii. This is achieved by decreasing the mixing-length (α) in my calculations, which inhibits the efficiency of convection and puffs up the stellar envelope, similar to the effect of strong magnetic fields (Chabrier, Gallardo & Baraffe 2007). Although I choose this method for modeling radius anomalies, the resulting pattern should be similar regardless of the inflation mechanism. I run the models to 120 Myrs, and display them in Fig. 3.1. The left panel quantifies the radius anomaly induced by each choice of α, as a function of mass. The red line represents the solar-calibrated

82 ∆R SSM ( R = 0, by deﬁnition), and the yellow, green, blue, and cyan lines represent calculations where the mixing-length is progressively lower in steps of 0.2. The largest anomalies for these models are ∼ 10-15% above 0.9M , and ∼ 5% for lower ⊙ masses, consistent with the range of observed anomalies (Stassun et al. 2012).

The right panel of Fig. 3.1 shows the resulting LDPs, plotted alongside the

Pleiades data from Fig. 1.2. These models were calculated with [Fe/H] = +0.03

(Table 1.1), and have been corrected for theoretical systematics using the method described in Section 2.4. In the absence of additional mixing mechanisms, the qualitative features of the Pleiades LDP are well reproduced by inﬂated models. The solar mixing-length LDP neatly traces the lower envelope of the data below ∼ 6000K, and the most inﬂated LDP closely brackets the upper envelope of the distribution.

Similar to the empirical data, the dispersion in the models is tight above 1 solar mass, and widens considerably towards cooler stars: ∆A(Li) = 0.21 dex at 6000K,

0.52 dex at 5500K, 1.18 dex at 5000K, and 2.83 dex at 4500K. Given errors in Teﬀ and A(Li), the upper envelope of the distribution appears somewhere between the models with α = 1.28 and 1.08. This corresponds to radius anomalies ∼ 2-4% at

0.6 M , 4-7% at 0.8 M , and 8-12% at 1.0 and 1.2 M . ⊙ ⊙ ⊙

While this qualitative agreement is excellent, there are a few additional factors to account for. First, the correlation between rotation rate and radius anomaly suggests that stars converge to their standard model radii as they spin down on the

MS. When the initially inﬂated models deﬂate to their solar-calibrated radii, the

83 surface temperature will increase, and the stars near the top of the Li distribution will move towards the left in my plots. This may reduce the apparent Li dispersion in Fig. 3.1. Second, errors in the age and metallicity of the Pleiades will alter the location of the theoretical LDPs. To address these issues, I recast my models in Fig.

3.2. The lower blue line in each panel reflects the standard model LDP at the fiducial metallicity and age, and the upper blue line reflects inflated models calculated with the mixing-length α = α − 0.8. The shaded blue regions in the left panel ⊙ represent the range of locations these bounds could lie in, given the quoted 1σ [Fe/H] errors. The predicted bounds still neatly bracket the available data, confirming the qualitative suggestion of Fig. 3.1. The central panel shows two additional upper and lower LDPs, calculated for 1σ age errors. There is no substantial difference between the theoretical prediction at these three ages, suggesting that the relevant

Pleiades stars have entered the MS, and no longer change on Myr timescales. Finally, the right panel of Fig. 3.2 shows an LDP that was inflated on the pre-MS, but whose radii have converged to standard predictions (red line). SSM Li depletion is completed before main sequence spin-down commences in all stars, so the Li abundance is not affected when stars deflate. The LDP therefore shifts only in Teff , and not in A(Li), as represented by the red arrows. The deflated models still trace the upper envelope of the Pleiades well, confirming that the cool star Li dispersion persists after substantial main sequence spin-down has occurred. Whether this has

84 occurred yet is unclear, but the distribution width is large enough to explain the data in either case.

With these eﬀects included, my models self-consistently explain many features of the Pleiades LDP: 1) the small dispersion in hot stars, 2) the median abundance of hot stars, 3) the dispersion in cool stars, 4) the locations of the upper and lower envelopes of the cool star distribution, and 5) the rotation-Li correlation seen below 5500K. Given that these models were calibrated only to reproduce the empirically-observed radius anomaly, without regard for the Li predictions, the excellent agreement between data and theory strongly suggests that pre-MS inﬂation is the cause for the Pleiades Li dispersion.

3.4. Additional Clusters

As a test of the generality of this picture, I extend this full analysis to six additional young clusters: NGC 2264, β Pictoris, IC 2602, NGC 2451 A+B, α Persei, and

Blanco 1. Eﬀective temperatures and A(Li)s were taken directly from the literature for these clusters. Ages and [Fe/H]s were obtained from various sources, for calculating their respective model predictions. These sources are listed in Table 3.1.

NGC 2451 A and NGC 2451 B are two diﬀerent clusters along the same line of sight, but since they appear to have similar ages and compositions, I combine their data into a single set. Each age comes from the LDB technique, except that of NGC 2451

A+B, which comes from ﬁtting isochrones to the main sequence turn-oﬀ, and that

85 of NGC 2264. The age of NGC 2264 is a contentious topic; previous studies place it between 0.1 Myr and 10 Myr (see Dahm 2008 for a thorough discussion), but most authors agree there is a substantial age spread within the cluster population. I adopt the age of 6 ± 3 Myr, to roughly bracket the range of literature ages. Each quoted

Fe abundance was measured with high-resolution spectroscopy, though I caution that they were not derived uniformly. Furthermore, each author employed their own methodology for deriving eﬀective temperatures and Li abundances. I consider this level of precision acceptable, since these clusters will be used to seek qualitative agreement rather than quantitative rigor.

These data are shown in Figs. 3.3 and 3.4, along with models corresponding to the quoted age and metallicity of each cluster. The meaning of the ﬁgures in each row is the same as in Fig. 3.2. Each has been corrected for theoretical systematics by scaling relative to the calibrated Pleiades LDP described in Section 2.4, except

NGC 2264, which has not yet undergone enough Li depletion for my systematic corrections to be meaningful.

The agreement between the data and models is good for each cluster. The dispersion present in the Pleiades is clearly a general phenomenon which develops between 6 and 20 Myr, and persists onto the ZAMS. This is consistent with an early origin of the Li spread, which my models predict develops between 8 and 15

Myr, and the subsequent termination of additional depletion until main sequence mechanisms kick in. There is a dip present at 4000K in NGC 2264, which agrees

86 well with the non-inflated models. At this same temperature, there are no data points that are consistent with the inflated upper envelope. This could reflect a time delay in the development of the dispersion: at this young age, proto-stars are still contracting and spinning up, and therefore may not yet possess their ultimate radius anomaly. In the left columns, each cluster older than NGC 2264 falls within the bounds of my models for a < 1σ [Fe/H] value. The Pleiades, β Pictoris, NGC

2451, and Blanco 1 are all fit well by models generated with the fiducial [Fe/H], but α Per is best fit by a metallicity close to solar. This agrees well with the α

Per iron abundance measured by Boesgaard & Friel (1990), who found an [Fe/H] nearly identical to that of the near-solar Pleiades. The only clusters whose lower distribution is substantially diﬀerent from the models is IC 2602, which appears to lie a few 100K to the right of the predicted lower bound. This could be accounted for by assuming a slightly younger age for the cluster (see below).

The central columns shows that the early-time LDP is quite sensitive to age.

Li patterns rapidly evolve both in A(Li) and in Teﬀ during their early stages, before stabilizing around 100 Myr. Each cluster appears consistent with its ﬁducial age, except IC 2602, whose LDP appears closer to 40 Myr old than 46 Myr old, in agreement with a recent LDB reanalysis (Soderblom et al. 2014). This suggests that if the composition of a cluster is well known, and the distribution of pre-MS radii is taken into account, LDPs may be a viable method for inferring young cluster ages. This possibility has been doubted in the past, as the cause of the Li dispersion

87 was not understood (i.e. Jeﬀries et al. 2009). However, these ﬁgures suggest that strong age constraints can be obtained through comparison with theory, even with my qualitative models. Furthermore, young cluster Li patterns are a useful way to measure the ages of clusters in the range 5-15 Myr, when my models predict that the dispersion develops. Bell et al. (2013) have recently proposed a revised age scale for several clusters younger than 20 Myr, which suggested new ages approximately double their previous literature values. Li observations could strongly discriminate between their new ages and previous ages in several cases. For example, Bell et al. (2013) found an age of 12 Myr for NGC 2362, up from 6.3 Myr (D’Antona &

Mazzitelli 1997). The former age implies signiﬁcant depletion in some members, similar to the pattern of β Pictoris, while the latter suggests only mild depletion and small dispersion, similar to the pattern of NGC 2264. This could provide a strong test of the age scale of young clusters.

Finally, the right columns shows that even after the radius anomaly has vanished, the dispersion persists in all clusters older than 10 Myr. The red line presents a superior qualitative representation of the upper envelope of NGC

2451(A+B), α Per and Blanco 1, suggesting that, by 65 Myr, stars have spun down enough to suppress their radius anomalies. These clusters may be in transition from an inﬂated to a deﬂated upper envelope. By contrast, the red line is clearly a worse

ﬁt to the upper envelopes of β Pic and IC 2602 than the blue line, implying that the most inﬂated stars are still rotating quite rapidly at these young ages. The general

88 picture that arises from this plot is that stars develop a Li dispersion around 10

Myr due to a large range in radii at ﬁxed mass, and converge on SSM predictions of radius and Teﬀ sometime between 50 and 100 Myr.

It is worth re-emphasizing that these models were not calibrated to reproduce the Li patterns seen in Figs. 3.1-3.4. The mixing-lengths used in my inﬂated models were chosen to qualitatively reproduce the range of radius anomalies seen in rapidly rotating systems. This implies that the Li patterns presented above are purely predictions of the model, not calibration points. The success of these models in reproducing the depletion patterns of many young and ZAMS clusters is very encouraging, and strongly implicates radius anomalies as a key ingredient in shaping young, cool star abundances. I note that the upper envelopes in Figs. 3.3 and

3.4 represent near-maximal inflation, and thus are almost certainly an optimistic prediction of the true upper envelope of the distribution. However, as described in the beginning of this chapter, several observational effects may artificially increase the measured Li abundance in these stars (e.g. line formation physics, rapid rotation, spots). If one were able to deconvolve the actual abundances from errors brought on by observational effects and Poisson noise, the maximum radius anomaly needed to reflect the upper envelope would likely decrease. Therefore, even in the event that

10-15% is an optimistic upper limit, this eﬀect may produce a suﬃcient abundance spread.

89 It is also worth noting that several open clusters contain a population of Teﬀ ∼

5500-6000K stars that lie signiﬁcantly below the mean trend. These can be seen in the Pleiades in Fig. 3.2, NGC 2451 A+B and α Per in Fig. 3.4, the Hyades in Fig.

1.2, and in many other clusters (i.e. IC 4665 − Jeﬀries et al. 2009; NGC 3532 −

Steinhauer 2003). These stars cannot be explained through depletion mechanisms considered in this work, since the one non-standard effect I have included can only suppress Li depletion. Non-member contamination cannot be completely ruled out, even when objects are consistent with membership, which may explain a portion of these objects. However, if some are indeed cluster members, additional physical processes must have influenced their depletion history, the exploration of which is beyond the scope of this paper. I therefore call attention to these objects as possibly reflecting a distinct Li depletion pathway, but do not speculate about their nature. Further observational work, particularly a robust confirmation of cluster membership, should be undertaken to investigate these objects.

3.5. Consequences of the Radius Anomaly in Young Clusters

The presence of radius anomalies in young systems has consequences for some astrophysical measurements. Fig. 3.5 shows pre-MS tracks and isochrones for solar calibrated SSMs. The solid red triangle represents the HR diagram location of a

0.9 M , 3 Myr old model with standard radius, and the faded red triangle represents ⊙ the HR diagram location of a star with the same mass and age, but with a radius

90 anomaly of ∼ 10%. As can be seen, this eﬀect shift pre-MS isochrones towards cooler temperatures and younger ages. The result is that while the faded triangle is

0.9 M and 3 Myr old, isochrone matching would infer a mass of ∼ 0.65 M and ⊙ ⊙ an age of ∼ 1.7 Myr, resulting in errors of ∼ 30% and 40% respectively for a 10% radius anomaly. This is an important eﬀect, since the fundamental parameters of young stars are routinely measured by placement on a grid of pre-MS isochrones.

If this method is used to derive the masses and ages of individual stars, inﬂated objects will be systematically older and more massive than inferred. This could have important consequences for several calculations. First, many studies use masses derived for pre-MS stars to measure the initial mass function (IMF) of the stellar distribution. The presence of inﬂated stars will tend to scatter individual mass measurements towards low values relative to the true mass, causing the inferred

IMFs to be bottom-heavy. Other studies use the locus of pre-MS stars on the HR diagram to estimate the age of clusters. If a range of radius anomalies is present, the data will not fall on a single isochrone, but instead be spread out between a standard isochrone and a maximally inﬂated isochrone. This can induce age errors up to 40%.

Inﬂated radii may also bias cluster age measurements obtained through the

LDB technique by altering the central temperature of FCSs, and thus the age at which they deplete Li. To test this, I run standard and inflated stellar models of fully convective stars to the ZAMS. I find that if a star is inflated on the pre-MS, the onset of Li depletion will be delayed compared to a SSM star of equal mass, and

91 thus systematically bias ages to younger values. In particular, if a range of radius anomalies are present in a young cluster, the transition from Li-rich to Li-poor will be fuzzed out over a characteristic mass range, since the transition mass at fixed age depends on the degree of inflation. My findings are consistent with the studies of Burke et al. (2004) and Macdonald & Mullan (2010). However, they are not consistent with Yee & Jensen (2010), who argue that pre-MS inflation may lead to

LDB estimates that are older than the true ages. They reach this conclusion by correctly noting that inflated stars have a lower Teff , which causes FCS masses to be systematically underestimated by SSM isochrones. This will in turn lower the inferred transition mass, leading to an older inferred age. However, their analysis assumed that the age at which Li destruction occurs is not altered by inflation, which my models rule out. The effects of inflation on individual bandpass magnitudes and bolometric corrections may further complicate matters, and deserve further investigation. Finally, I note that I have used LDB ages for the clusters in this section. If these ages are incorrect, it could impact the agreement between open cluster data and my models. However, I find that a reduction of α of 0.8 produces a maximum age error of ∼10% which, given the minor sensitivity of LDPs to age

(center columns, Figs. 3.2-3.4), does not change the qualitative conclusions of this section.

92 3.6. Summary

I have proposed a explanation for the cool star Li abundance spread present in young clusters. This dispersion would arise naturally in the presence of a radius dispersion at fixed mass during the pre-MS. Interferometric observations of single stars and precision tests of stellar models using detached eclipsing binaries evince the existence of inflated radii in some chromospherically active stars. Activity also correlates with rotation, so the most rapidly rotating stars likely possess the largest radius anomalies. If these stars are inflated on the pre-MS, the temperature at the base of their CZs will be less than their slowly rotating counterparts, strongly inhibiting the rate of pre-MS Li depletion, and causing them to be over-abundant in

Li on the ZAMS.

I imposed a radius anomaly on my models of ∼ 5-15%, consistent with the upper envelope of observed anomalies, by reducing the eﬃciency of convection. Li depletion is dramatically inhibited in these models, leading to a > 1 dex spread at some temperatures. These models succeeded in qualitatively predicting the upper envelope of the Li distribution of the Pleiades. Furthermore, this theory naturally predicts the rotation-abundance correlation seen in cool Pleiads. I then extended my models to 6 additional young clusters, and found good agreement between the empirical evolution of Li patterns from 0-120 Myr and my models. Since the inﬂated models were not calibrated to reproduce the Li patterns of these clusters, and instead were calibrated to match the observed upper envelope of radius anomalies in rapidly

93 rotating systems, their success in reproducing the LDPs of several young clusters strongly supports the validity of this theory. Finally, I ended with a discussion of some eﬀects that inﬂated radii could have on pre-MS isochrones, stellar IMF measurements, and ages determined through the LDB technique.

Improved measurements of Li and rotation in clusters shortly before and after the epoch of SSM burning will provide a crucial test of this theory. Equal mass stars with different rotation rates will show similar Li abundances before 5 Myr, and radically different abundances by ∼ 15 Myr, with the slower rotating stars showing larger degrees of depletion. The 13 Myr old cluster h Per is a prime target for studying the late-time dispersion evolution, as a rich rotation data set has recently become available (Moraux et al. 2013). Fast and slow rotating stars of equal mass can be targeted, providing a measure of both the Li spread at fixed Teff , and of the correlation between Li abundance and rotation at 13 Myr.

I now turn to the next step in solidifying the predictions of this theory.

In particular, I attempt to explicitly tie the radius anomaly to stellar rotation rates through empirical correlations (Stassun et al. 2012), and account for the simultaneous effects of inflation and rotational mixing. In the next chapter, I will present calculations of inflated rotating models, and argue that they are successful in reproducing the qualitative shape of the Pleiades Li and rotation distribution.

94 4.0 α = 1.88 15 α = 1.68 α = 1.48 Low α α = 1.28 3.0 α = 1.08

10 2.0

Solar α A(Li) (%) R R ∆ 1.0

0.0

0 −1.0 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 6500 6000 5500 5000 4500 4000 Mass (M ⊙) Effective Temperature (K)

Fig. 3.1.— Impact of radius inflation on lithium depletion at 120 Myr. The radius anomalies and Li abundances resulting from a decreased mixing length on the pre-MS. The left panel shows the fractional change in the radius of stellar models with reduced mixing-lengths, relative to the solar-calibrated mixing-length model (red line), as a function of mass. Radius anomalies of ∼ 5-15% on the pre-MS can inhibit SSM depletion enough to account for most cool stars in the Pleiades, as seen in the right panel. This paradigm predicts a large spread developing at fixed effective temperature around 10 Myr.

95 Metallicity Errors Age Errors Radius Anomaly 3 3

2 2

1 1 A(Li)

0 0

Age = 117 Myr −1 −1 Pleiades Age = 125 Myr Upper/Lower Bounds [Fe/H] = 0.03 ±0.02 Age = 133 Myr Deflated Models Age = 125 ±8 Myrs −2 −2 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 3.2.— Cluster parameter errors and inflated radius lithium depletion (1/3). The effects of three sources of error on the agreement between the Li predictions of inflated stellar models and the empirical Pleiades LDP. Left: Blue lines represent inflated (upper) and standard (lower) models calculated with the best-guess cluster parameters for the Pleiades. The blue shaded bands represent the locations these models could lie given 1σ [Fe/H] errors. Center: Blue lines again represent best- guess Pleiades models. The red dashed and green dotted lines represent the locations of the models given 1σ age errors. Right: Blue lines represent the best guess models for the Pleiades, and the red dashed line represents the location of the upper envelope, once the models have deflated to their standard radius predictions.

96 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K) 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500

Metallicity Errors Age Errors Radius Anomaly 3 3

2 2 A(Li) 1 1 A(Li)

0 0

Age = 3 Myr −1 NGC 2264 −1 Age = 6 Myr Upper/Lower Bounds [Fe/H] = -0.15 ±0.05 Age = 9 Myr Age = 6 ±3 Myrs Deflated Models −2 −2 Metallicity Errors Age Errors Radius Anomaly 3 3

2 2 A(Li) 1 1 A(Li)

0 0

−1 β Pictoris Age = 17 Myr −1 Age = 21 Myr Upper/Lower Bounds [Fe/H] = -0.01 ±0.08 Age = 25 Myr Age = 21 ±4 Myrs Deflated Models −2 −2 Metallicity Errors Age Errors Radius Anomaly 3 3

2 2 A(Li) 1 1 A(Li)

0 0

Age = 40 Myr −1 IC 2602 −1 Age = 46 Myr Upper/Lower Bounds [Fe/H] = 0.00 ±0.01 Age = 52 Myr Age = 46 ±6 Myrs Deflated Models −2 −2 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 3.3.— Cluster parameter errors and inﬂated radius lithium depletion (2/3). The same as in Fig. 3.2 for NGC 2264 (top), β Pictoris (middle), and IC 2602 (bottom).

97 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K) 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500

Metallicity Errors Age Errors Radius Anomaly 3 3

2 2 A(Li) 1 1 A(Li)

0 0

Age = 50 Myr −1 NGC 2451(A+B) −1 Age = 65 Myr Upper/Lower Bounds [Fe/H] = -0.01 ±0.08 Age = 80 Myr Age = 65 ±15 Myrs Deflated Models −2 −2 Metallicity Errors Age Errors Radius Anomaly 3 3

2 2 A(Li) 1 1 A(Li)

0 0

Age = 80 Myr −1 α Persei −1 Age = 90 Myr Upper/Lower Bounds [Fe/H] = -0.05 ±0.05 Age = 100 Myr Age = 90 ±10 Myrs Deflated Models −2 −2 Metallicity Errors Age Errors Radius Anomaly 3 3

2 2 A(Li) 1 1 A(Li)

0 0

Age = 108 Myr −1 Blanco 1 −1 Age = 132 Myr Upper/Lower Bounds [Fe/H] = 0.04 ±0.02 Age = 156 Myr Age = 132 ±24 Myrs Deflated Models −2 −2 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500 6000 5500 5000 4500 4000 3500 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 3.4.— Cluster parameter errors and inﬂated radius lithium depletion (3/3). The same as in Fig. 3.2 for NGC 2451 A+B (top), α Persei (middle), and Blanco 1 (bottom).

98 1.1M 1.0 ⊙ 0.5M ⊙

0.5 1 Myr

⊙ 20 Myr L L 0.0 5 Myr Log Log

−0.5

−1.0

−1.5 6000 5500 5000 4500 4000 3500 Effective Temperature (K)

Fig. 3.5.— Impact of radius inflation on luminosity and temperature. The impact of the radius anomaly on the location of pre-MS stars on the HR diagram. Black dotted lines represent the location of SSM evolutionary tracks for M = 1.1-0.5M , and the ⊙ black solid lines show SSM isochrones at 3 ages. The solid red triangle shows the HR diagram location of a 0.9M , 3 Myr old SSM, while the faded triangle shows the ⊙ same model with a radius inflation of ∼ 10%. Underestimates of 30% and 40% in the mass and age would result if the stellar parameters of the inflated star were inferred with these standard pre-MS models.

99 Cluster A(Li)/Teﬀ Age [Fe/H] Parameter Sources Source1 (Myr) (Age, [Fe/H])a

NGC 2264 (1) 6 ± 3 -0.15 (2), (3) β Pictoris (4) 21 ± 4 0.01 ± 0.08 (5), (6) IC 2602 (7) 46 ± 6 0.00 ± 0.01 (8), (9) NGC 2451 A+B (10) 65 ± 15 -0.01 ± 0.08 (11), (11) α Persei (12) 90 ± 10 -0.05 ± 0.05 (13), (14) Blanco 1 (15) 132 ± 24 0.04 ± 0.02 (16), (17)

1Sources: (1) Soderblom et al. 1999; (2) King et al. 2000; (3) Dahm 2008; (4) Torres et al. 2006; (5) Binks & Jeffries 2014; (6) Viana Almeida et al. 2009; (7) Randich et al. 2001; (8) Dobbie et al. 2010; (9) D’Orazi & Randich 2009; (10) Hünsch et al. 2004; (11) Hünsch et al. 2003; (12) Balachandran et al. 2011; (13) Stauffer et al. 1999; (14) Boesgaard & Friel 1990; (15) Jeffries & James 1999; (16) Cargile et al. 2010; (17) Ford et al. 2005

Table 3.1. Additional Cluster Data

100 Chapter 4: Rotation, Inﬂation, and Lithium in the Pleiades

4.1. Background

The classic Vogt–Russell theorem states that the mass and composition of a star uniquely determines its structure in hydrostatic equilibrium. Although this paradigm has proven to be a strong overall guide during the main sequence (MS), the implicit assumptions in the Vogt–Russell theorem break down in the pre-MS, where substantial star-to-star variations in luminosity are ubiquitous at fixed Teff in star forming regions (e.g. Hillenbrand 1997). This behaviour has traditionally been interpreted as the signature of an intracluster age spread, but an alternative explanation is that stars at fixed mass and age can have non-uniform stellar parameters (i.e. radius) induced by processes neglected in traditional models, such as rotation, mass accretion, and magnetic fields. Much attention has been paid to understanding these processes, as a complete picture of pre-MS evolution is crucial for inferring the ages of young star forming regions, measuring the low end of the stellar initial mass function, and constraining the distribution of circum-stellar disc lifetimes, which in turn have important implications for both the angular momentum evolution of stars and the formation of planets.

101 The light element 7Li is a sensitive diagnostic of pre-MS structure. Lithium

7 6 burns through the Li(p, α)α reaction at a temperature TLi ∼ 2.5 × 10 K, and is therefore directly destroyed in stellar envelopes when the temperature at the base of the surface convection zone (TBCZ) is high. During the period of deep convection on the pre-MS, TBCZ surpasses TLi for FGK stars, inducing Li depletion in the envelope

(Iben 1965). This ‘standard model’ depletion terminates during the approach to the zero-age main sequence (ZAMS), when the convection zone retreats and again becomes cool. Because the rate of Li burning scales as T 20 near the destruction temperature (Bildsten et al. 1997), the magnitude of standard model depletion is extremely sensitive to TBCZ. In standard stellar models (SSMs), only mass and composition inﬂuence this temperature, so equal mass stars within clusters are expected to reach the ZAMS with equal surface Li abundances, and remain nearly

ﬁxed in A(Li)1 until the main sequence turn-oﬀ.

By contrast, Li abundance spreads have been observed in many stellar populations, such as the late G and K dwarfs on the pre-MS and ZAMS (e.g. Somers

& Pinsonneault 2014, and references therein), the mid-F dwarfs of 0.1–0.5 Gyr old clusters (e.g. Boesgaard & Tripicco 1986; Balachandran 1995), and solar analogues of old clusters and the ﬁeld (Sestito & Randich 2005 and references therein; Israelian et al. 2009). Various non-standard processes have been put forward to explain the development of these dispersions, but they are generally unable to simultaneously

1 A(Li) = 12 + log10(n(Li)/n(H))

102 explain all three above classes. For example, rotational mixing can induce depletion and dispersion on the MS, but cannot replicate the pattern in young stars (see below). Explaining this multifaceted Li depletion pattern presents a serious challenge for stellar evolution theory, but its resolution must reveal interesting physics not included in the standard model.

In this chapter, I investigate a potential cause of the Li dispersion developing on the pre-MS. This dispersion has been identiﬁed in several young clusters, such as

Blanco 1 and α Per (Jeﬀries & James 1999; Balachandran et al. 2011), but I focus solely on the most notable example, the Pleiades. The Pleiades is a nearby open cluster (d ∼ 133 pc; An et al. 2007), just beyond the K dwarf ZAMS (t ∼ 125 Myr old; Stauﬀer, Schultz & Kirkpatrick 1998), and with a near-solar iron abundance

([Fe/H] ∼ 0.03 ± 0.02; Soderblom et al. 2009). The Pleiades also exhibits a significant, Teff dependent spread in the λ6708 Li I absorption feature, originally identified by Duncan & Jones (1983) and later confirmed by Soderblom et al. (1993a).

This EW dispersion suggests an abundance spread at ﬁxed Teﬀ of a few tenths of a dex at 6000K, which increases to greater than an order of magnitude below 5500K

(Fig. 4.1). This dispersion has remained unexplained since its discovery, but an intriguing clue has motivated several attempts at resolution: the most Li rich stars at ﬁxed Teﬀ tend to be the most rapidly rotating in the K dwarf regime (Soderblom et al. 1993a). This observation has led several authors to suggest that the dispersion results from various observational biases related to rapid rotation, such as the

103 enhancement of the λ6708 Li I feature due to blending from rotational broadening

(Margheim et al. 2002), large spot filling factors and chromospheric activity altering observed equivalent widths (Soderblom et al. 1993a), and non-LTE effects (Carlsson et al. 1994). However, King et al. (2010) have shown that while these effects may contribute to the total dispersion, some underlying Li spread is likely present (see the discussion in section 6.2 of Somers & Pinsonneault (2014)). It thus appears that some mechanism related to rotation affects Li depletion on the pre-MS. As discussed in §1.5, the strongest candidate at present is rotational mixing.

While rotational mixing may be responsible for main sequence depletion, it is unlikely to produce the pattern seen in the Pleiades K dwarfs. Li depletion through rotational mixing is most eﬃcient in rapid rotators, opposite to the sense of the

Li–rotation correlation in the Pleiades. This implies that some stronger process related to rotation on the pre-MS must counteract the depletion of Li through mixing. Several such mechanisms have been suggested in the literature, such as stronger internal shears in slow rotators due to disparate core–envelope coupling times (Bouvier 2008), enhanced Li depletion in stars with extreme accretion histories

(Baraﬀe & Chabrier 2010), and enhanced internal shears due to long disc-locking times in slow rotators (Eggenberger et al. 2012). I note that each of these mechanisms induces a Li dispersion by enhancing the depletion of slow rotators, whereas the empirical calibration of Somers & Pinsonneault (2014) favoured the suppression of depletion in rapid rotators.

104 In chapter 3, I proposed an explanation for the Pleiades Li spread: a dispersion in stellar radii at fixed mass and age during the pre-MS. If the convective envelope of a star were inflated relative to standard predictions, the temperature at the base of the convection zone would be reduced. Li destruction in the envelope is a strong function of this temperature, so stars of dissimilar radii would undergo different rates of Li depletion during the pre-MS, and thus arrive at the ZAMS with unequal Li abundances. This effect is so strong that a pre-MS radius spread of ∼10 per cent in non-rotating models produces a greater than order-of-magnitude spread in Li at the

ZAMS (Fig. 3.1), and if the most inﬂated stars were also the most rapidly rotating, the observed Li–rotation correlation would be naturally produced.

This idea was motivated by the rapidly-growing literature on discrepancies between the precisely measured radii of stars and the predictions of modern stellar evolution codes. Accurate radius measurements (∼1–3 per cent) can be obtained by analysing the light curves of detached eclipsing binaries (e.g. Popper 1997; Torres,

Anderson, & Giménez 2010), and by measuring the angular diameter of nearby stars with interferometry (e.g. Boyajian et al. 2012). The average radii of open clusters can also be inferred by statistically modelling projected radius measurements (R sin i; Jackson, Jeffries & Maxted 2009; Jackson & Jeffries 2014a). With each method, discrepancies between predicted and observed radii of ∼10 per cent have been found.

These so-called ‘radius anomalies’ have been identiﬁed, and found to correlate with rotation, in the pre-MS cluster IC 348 (Cottaar et al. 2014). They have also been

105 claimed in young main sequence clusters, including the Pleiades (Jackson & Jeﬀries

2014a), and the magnitude of the discrepancy has been seen to correlate with chromospheric activity and rotation rate (L´opez-Morales 2007; Clausen et al. 2009;

Stassun et al. 2012, 2014).

In Chapter 3, I showed that a ∼10 per cent radius dispersion on the pre-MS can produce a Li dispersion commensurate with the Pleiades pattern. However, the impact of rotational mixing was not included. Mixing-driven Li destruction is strongest in rapid rotators, so this effect could compete with the inflationary inhibition of Li destruction. I therefore seek to assess whether a connection between rotation and radius inflation during the pre-MS represents a viable class of solutions to the Pleiades Li problem. To address this question, I simultaneously calculate the effects of rotational mixing and radius inflation on Li depletion in my evolutionary models, and compare the results to the empirical Pleiades distribution. To model radius inflation, I apply a modification to the mixing length in my stellar evolution code (Section 4.2.1). This approach provides us with a generic framework within which to explore the impact of radius on Li destruction. I construct a grid of models with dimensions of mass, radius inflation, and rotation using this method (Section

4.2.2), and collect a catalog of Pleiades data, for comparison with my models

(Section 4.2.3).

With this model grid, I explore the individual and simultaneous impacts of rotation and radius inﬂation on Li abundances (Section 4.3), and conclude that large

106 inﬂation factors can indeed suppress rotationally-driven Li destruction in stellar models. Based on this result, I explore whether a putative connection between the two properties can produce a pattern which resembles the empirical Pleiades distribution (Section 4.4). Several limiting cases are ruled out, and a special case where faster rotation leads to greater radius inﬂation is found to compare favourably with the Pleiades. Thus, I conclude that the important qualitative features of the

Pleiades Li and rotation pattern can be predicted in stellar models if and only if radius inﬂation is correlated with rotation during the pre-MS. Detailed models tied to the underlying physical mechanism might well diﬀer in their behavior; I discuss such models in Section 4.5.1. Finally, I suggest an observational test for the presence of a radius dispersion in the Pleiades and other clusters in Section 4.5.2, and summarise my conclusions in Section 4.6.

4.2. Methods

The ﬁrst step in my investigation is to generate a grid of stellar models with dimensions of mass, radius anomaly, and initial rotation conditions. Once complete,

I can explore the generic behaviour of the models, and determine their ability to reproduce the Li–rotation correlation seen in the Pleiades. In this section, I ﬁrst describe the stellar evolution code I employ, my basic calibration procedures, and my treatment of radius inﬂation. Next, I describe the implementation of rotation in

107 my models. Finally, I assemble literature data for the Pleiades, as an empirical test of my models.

4.2.1. Standard model calibration and radius inﬂation

Stellar models were calculated using the Yale Rotating Evolution Code (YREC; see

Pinsonneault et al. 1989 for a discussion of the mechanics of the code). I adopt a present day solar heavy element abundance of Z/X = 0.02292 from Grevesse &

Sauval (1998), and choose the hydrogen mass fraction (X) and the mixing length coeﬃcient (αML) that reproduce the solar luminosity and radius for a 1M model at ⊙ 4.568 Gyr. The ﬁnal solar calibrated values are X = 0.71304, Z = 0.018035, and α ⊙ = 1.88166. Our models use the 2006 OPAL equation of state (Rogers, Swenson &

Iglesias 1996; Rogers & Nayfonov 2002), atmospheric initial conditions from Kurucz

(1979), high temperature opacities from the opacity project (Mendoza et al. 2007), low temperature opacities from Ferguson et al. (2005), the 7Li(p, α)α cross section of

Lamia et al. (2012), and the cross-sections of other nuclear reactions from Adelberger et al. (2011). I treat electron screening using the method of Salpeter (1954), and do not include diﬀusion, as the timescales of this process are very long compared to the ages of systems I are concerned with. Our models are initialised with a Li abundance equal to the proto-solar abundance A(Li) = 3.31 (Anders & Grevesse 1989).

I model radius inﬂation in this paper by reducing the mixing length parameter

αML, as suggested by Chabrier, Gallardo & Baraﬀe (2007). While inhibited

108 convection can result directly from the presence of magnetic fields in the convective envelope of stars, I do not ascribe a specific underlying cause to the radius inflation considered in this paper. Due to the adiabatic nature of convective regions, a given fractional increase in the radius will have a similar effect on the temperature at the base of the envelope regardless of the underlying cause. I can therefore use an altered mixing length to explore the generic effects on Li abundances of any mechanism which increases the radii of stars. The mixing length is held fixed for each individual model during its lifetime, and is determined by requiring a specified inflation percentage at 10 Myr, an age characteristic of pre-MS Li destruction. This formulation embeds an age dependence of the radius anomaly in my models – in particular, the anomaly remains nearly fixed as a function of age, even while stellar rotation rates evolve. This simplification may not capture important details of inflation/rotation connections tied to specific physical mechanisms, which require more sophisticated modeling to uncover. However, this framework is appropriate for investigating the general consequences of radius inflation on Li abundances.

Using the solar calibrated mixing length α , I calculated a baseline set of ⊙ non-rotating models, with masses varying from 0.7M to 1.2M in steps of 0.05M , ⊙ ⊙ ⊙ corresponding to the range of Pleiades members considered in this paper. These models are treated as possessing a radius anomaly of 0 per cent. I then deﬁne the function δR(M,αML) as the fractional diﬀerence between the solar calibrated radius,

109 R(M,α ), and the radius of another model of equal mass but mixing length αML. ⊙

R(M, αML) − R(M, α ) δ (M, α )= ⊙ . (4.1) R ML R(M, α ) ⊙ This value is calculated at 10 Myr as described above, an age characteristic of the epoch of pre-MS Li depletion in solar analogues. Next, I determined the αML necessary to inﬂate a model of each mass to δR(M,αML) = 0.02, 0.04, 0.06, 0.08, and

0.10, corresponding to radius anomalies of 2–10 per cent. The required αML for each combination of mass and radius anomaly considered in this paper is shown in Fig.

4.1. A maximum anomaly of 10 per cent was chosen to coincide with the claimed radius anomalies in the Pleiades (Jackson & Jeﬀries 2014a) and the upper envelope of anomalies observed in detached eclipsing binary systems and interferometric targets (see Section 4.1).

I note that pre-MS Li depletion in stellar models is highly sensitive to the input physics. Somers & Pinsonneault (2014) showed that adopting different published physical inputs, such as the equation of state or the 7Li(p, α)α reaction cross-section, can produce significant changes in the expected pre-MS depletion of stars 0.9M ⊙ and below. However, Li depletion on the main sequence is insensitive to physical inputs, so main sequence depletion predictions are robust once pre-MS depletion is correctly predicted. To address this issue, Somers & Pinsonneault (2014) performed an empirical calibration of their models using the warm (Teff > 5500K) stars in the

Pleiades (see Sections 3.3–3.4 for details), and derived systematic corrections to the

ZAMS Li abundance as a function of Teﬀ . The corrections range from ∼0.2 dex

110 in A(Li) at 6000K to ∼1 dex at 5000K. In this work, I apply these ‘empirically calibrated’ pre-MS Li depletion corrections to my models.

4.2.2. Rotation and angular momentum evolution

To initialise rotation, I first evolve non-rotating models down the Hayashi track to the deuterium birth line, defined as the age at which 1 per cent of the initial D abundance has been destroyed (∼ 0.1 Myr for the Sun). At this age, the launch conditions are applied. I assign an initial rotation period Pi, and assume this period to be fixed for a specified time τD, though magnetic interactions with a circum-stellar disc (e.g. Koenigl 1991). Longer disc-locking times lead to slower subsequent rotation rates, as the stars are prevented from spinning up until a later epoch. Thereafter the rotation rate evolves under the influence of AM loss at the surface through magnetised winds. To model the loss of angular momentum, I adopt a modification of the wind law of Kawaler (1988), first proposed by Krishnamurthi et al. (1997). In this formulation, the loss rate scales with the cube of the angular velocity, up to a critical ‘saturation’ velocity ωcrit, when the loss law becomes linear with ω. The saturation threshold ωcrit is scaled by the Rossby number (rotation frequency ω times the convective overturn time-scale τCZ ) of each model, which is normalised on the Sun (ω τCZ, ∼ 2.2). The formula is given by Eq. 4.2, ⊙ ⊙

1 1 2 2 2 R M − ωcrit τCZ  FK ω , ωcrit <ω dJ µR ¶ µM ¶ µ ω ¶ τCZ, =  ⊙ 1 ⊙ 1 ⊙ ⊙ (4.2)  2 2 2 dt  R M − ωτCZ τCZ FK ω , ωcrit ≥ ω µR ¶ µM ¶ µω τCZ, ¶ τCZ,  ⊙ ⊙ ⊙ ⊙ ⊙  111 FK represents the normalisation of the magnetic field strength, which is typically calibrated to reproduce observational data. Convection zones are assumed to rotate as a solid body, whereas radiative zones conserved angular momentum locally. The redistribution of angular momentum and chemical species throughout the radiative zones of star occurs purely through hydrodynamic instabilities in these models (e.g. Endal & Sofia 1978), and are calculated by solving the coupled differential equations,

I dω d I dω ρr2 = f ρr2 D , (4.3) M dt ω dr µ M dr ¶

dX d dX ρr2 i = f f ρr2D i . (4.4) dt c ω dr µ dr ¶

Here, I/M is the moment of inertia per unit mass, D is the characteristic diﬀusion coeﬃcient of the angular momentum redistribution processes (Pinsonneault et al. 1989; Pinsonneault, Deliyannis & Demarque 1992; Chaboyer, Demarque &

Pinsonneault 1995), fω is a scaling factor that sets the eﬃciency of this transport

(fω = 1 in my models), and fc is a scaling factor that sets the eﬃciency of transport of chemical species Xi relative to fω (Chaboyer & Zahn 1992).

Our rotation and mixing formulation leaves us with three free parameters: FK ,

ωcrit, , and fc. To calibrate the two rotation parameters, FK and ωcrit, , I adopt the ⊙ ⊙ standard technique of ﬁtting the median and rapid rotating envelopes of empirical distributions (e.g. van Saders & Pinsonneault 2013). As Li depletion begins as early

112 as a few Myr for higher mass stars, it is necessary to initialise my models near the birth-line. However, clusters around this age are unreliable rotation distributions benchmarks as many of their members retain their primordial circum-stellar discs, which can exchange angular momentum with their host stars in complicated ways.

Instead I select an older cluster, whose circum-stellar discs have dissipated, and solve for the range of initial conditions necessary to evolve towards its rotation distribution.

For this purpose, I choose the 13 Myr old cluster h Per (Moraux et al. 2013).

This cluster provides a useful benchmark for generating initial conditions as it lies on the pre-MS, and is old enough that gaseous accretion discs are no longer common.

I ﬁx the initial rotation period Pi = 8 days for all models, and ﬁnd that τD = 3

Myr accurately reproduces the median rotation period in h Per, and that τD = 0.1

Myr accurately reproduces the 90th percentile of rapid rotators, regardless of mass.

While an equal initial rotation period for all models may appear to be a crude approximation for an initial rotation distribution, there exist strong degeneracies in the choice of Pi and τD; in the absense of radius inﬂation, a fast spinning star with a long disk-locking time will be indistinguishable in both rotation and Li from a slowly spinning star with a short disk-locking time, once de-coupled from their disks

(e.g. Denissenkov et al. 2010). Even if diﬀerential radius anomalies (from 0-10 per cent) correlate with rotation before the age of h Per, I estimate that the resulting

Li abundance dispersion will be ∼0.2 dex at 13 Myr in the most extreme case, and

113 geneally much less. This dispersion is comparable to systemtics in the abundance determination, and far smaller than the Li dispersion my models seek to explain

(Section 4.4). I choose τD = 6 Myr as the upper limit for disc-locking times, in accordance with the observed upper limit of disc survival lifetimes (Haisch, Lada &

Lada 2001; Fedele et al. 2010).

With my launch conditions set, I calibrated FK and ωcrit, on the 550 Myr ⊙ old open cluster M37. I obtained the M37 rotation catalog from Hartman et al.

(2009), and determined both the median rotation rate of the full sample, and the median rotation rate of the most rapidly rotating 20 per cent, as a function of Teﬀ .

I then evolved stellar models using median and rapid launch conditions (τD = 3 and 0.1 Myr, respectively), with a variety of values for FK and ωcrit, , to the age of ⊙ the M37. The combination of parameters that best ﬁt the cluster rotation pattern are FK = 3.3 and ωcrit, = 9.5ω . These parameters are broadly consistent with ⊙ ⊙ values obtained by other authors (e.g. Krishnamurthi et al. 1997). With the rotation parameters set, fc was determined by requiring a median rotating solar mass model to match the median, metallicity-unbiased Li measurements of the Hyades from

Somers & Pinsonneault (2014), a cluster of similar age to M37. With this method, I obtain fc = 0.05. This value is consistent with typical results for rotational mixing calculations in the literature (e.g. Pinsonneault et al. 1990) and with anisotropic turbulence expected from theory (Chaboyer & Zahn 1992).

114 With the necessary values of αML in hand, and the angular momentum and chemical transport calibrated, I calculated a grid of Pi = 8 day, solar metallicity models, with dimensions of mass (0.7–1.2M ), δR (0.0–0.1), and τD (0.1–6 Myr). ⊙

4.2.3. Pleiades data

To construct a comparison data set of Teﬀ , A(Li), and Prot for the Pleiades, I draw

Teff s and A(Li)s from Somers & Pinsonneault (2014), and cross-correlate this sample with the Pleiades rotation catalog of Hartman et al. (2010). The resulting data can be seen in Fig. 4.1. I note that the Pleiades K dwarfs are known to show colour abnormalities possibly related to surface magnetic activity, radius inflation, and star spots (Jones 1972; van Leeuwen, Alphenaar & Meys 1987; Stauffer et al. 2003).

These eﬀects may also impact the measurement of the λ6707.8 Li I equivalent width, and thus the inferred abundance. However, these eﬀects have little impact on the forthcoming results, as I seek to understand only the generic features of the empirical pattern, which appear secure (see Section 4.1), and not the detailed temperatures or abundances of the individual stars.

4.3. The impact of rotation and inﬂation

Before performing a direct comparison between the models described in Section

4.2 and the empirical Pleiades pattern, I explore the generic impact of inﬂation and rotation on the destruction of Li during the pre-MS in the models. I begin by

115 reviewing the individual consequences of these two effects on stellar Li abundances in Section 4.3.1–4.3.2. Rapid rotation is shown to enhance the rate of Li depletion due to the strong influence of rotational mixing, and radius anomalies are shown to inhibit the rate of Li depletion due to a decreased temperature at the base of the convection zone. Next, I investigate models with both rotation and inflation in

Section 4.3.3. The resulting pattern is more complicated: Li depletion can either increase or decrease compared to a standard benchmark (no inflation, no rotation) depending on the relative strength of the two effects. In particular, I find that if a connection between rotation and inflation is present, Li abundance tends to correlate with rotation in cool stars. This motivates an exploration of the observational signature of different plausible inflation/rotation relationships in the next section.

Readers interested only in the ﬁnal results may proceed to Section 4.4.

4.3.1. Rotation

In the theory of rotational mixing, meridional circulation and internal shears transport material between layers, leading to a gradual dilution of the convective envelope with Li-depleted material from the deep interior. Stars are born with a range of initial rotation conditions, so the rate of mixing diﬀers from star to star in young systems, leading to the emergence of Li dispersions. Both the convective envelope and the radiative interior spin down over the course of 0.5–5 Gyr, eventually losing memory of their initial conditions, and asymptoting to rotation proﬁle that

116 depends on age but not early rotation rate (e.g. Krishnamurthi et al. 1997). This process equalises the rate of late-time Li depletion between stars with diﬀerent

AM histories, thus freezing out the intra-cluster Li dispersion. Throughout stellar lifetimes, the strength of mixing depends both on the absolute rotation rate of the star and on the degree of differential rotation within the star; this makes Li depletion efficient at young ages and inefficient at old ages, given the progressive spin-down of stars. This generic picture of main sequence rotational mixing has been long established, but I am concerned in this work with the effect of rotation on the pre-MS, which is often neglected in mixing studies due to its small assumed effect relative to standard model depletion. I therefore present an exploration of the impact on rotation on Li during the pre-MS.

The top left of Fig. 4.3 shows rotation tracks for several solar calibrated (δR

= 0.0), 0.9M models with different disc-locking times. The models remain at ⊙ constant rotation period (8 days) until detaching from their discs, when pre-MS contraction begins to increase their rotation rate. Once the models fully contract on to the ZAMS (∼ 50 Myr), they begin to spin down under the influence of magnetic winds. Shorter disc-locking times lead to faster rotation at the ZAMS, given an equal initial rotation rate. Thus the observational result that disc lifetimes differ from star to star naturally implies a >1 dex range of rotation periods at the ZAMS.

The bottom left panel follows the evolution of Li in the envelope of each model.

Three stages of pre-MS Li evolution are revealed: 1) no depletion for the ﬁrst few

117 Myr; 2) an epoch of depletion from ∼ 3–20 Myr; 3) very little depletion on the late pre-MS. Thereafter, rotational mixing reduces A(Li) on the MS. The second pre-MS phase represents the time when the temperature at the base of the convection zone surpasses the Li depletion temperature (see Section 4.1). The rate of Li destruction during this phase is a strong function of rotation, so depletion in rapid rotators is high. This is because rotational mixing is particularly eﬃcient when the material directly below the convection zone is highly depleted, as is the case when convection zones are deep. I have also plotted the Li track of a standard, non-rotating model

(black dashed line) for comparison. This model depletes nearly the same amount of Li during the pre-MS as the slow rotating models (τD ∼> 1 Myr), suggesting that very rapid rotation (∼< 1 day period) is necessary for an appreciable enhancement to the rate of pre-MS depletion.

The right column of Fig. 4.3 explores the mass dependence of this behaviour at the age of the Pleiades. In the top panel, each individual line represents a fixed disc-locking time, and the points signify mass steps of 0.05M , with the 0.9M locus ⊙ ⊙ indicated by a dotted line. A large range of rotation rates persist at 125 Myr within my mass range. The rotation dispersion is largest below ∼ 5700K, and decreases towards larger Teff . We also see that the structural effects of rapid rotation lead to a reduction of the surface temperature by ∼ 50–150K, independent of spot-induced changes. This results from the additional pressure support provided by rotation at the equator, which leads to a lower effective gravity, and hence a larger equilibrium

118 radius. The bottom right panel shows Li isochrones at the age of the Pleiades.

Lower mass stars undergo deeper convection during the pre-MS, and spend more time on the pre-MS, resulting in more rapid Li destruction over an extended period of time, and ultimately a lower abundance at 125 Myr. Variable rotation induces a

Li dispersion commensurate with the 0.9M spread at all masses in the considered ⊙ range, suggesting that the 0.9M Li tracks are representative of the behaviour of Li ⊙ destruction in all of my models. Thus, my rotating models predict that at the age of the Pleiades, slow rotators should cluster near the standard prediction (dashed black line), and fast rotators should be depleted below the standard prediction by ∼1.5 dex.

These results are necessarily dependent on the details of the stellar models, notably the treatment of angular momentum transport and loss. While I have assumed that angular momentum is redistributed purely by hydrodynamics and mechanics (Section 4.1), some authors have argued that empirical rotation trends are better ﬁt by models which include additional transport processes operating in the interior (e.g. Bouvier 2008; Denissenkov et al. 2010; Gallet & Bouvier 2013), which could have implications for pre-MS Li destruction. A full treatment of this additional physics is outside the scope of this project, but I note that since rotational mixing can only destroy Li, and not inhibit its destruction, the non-rotating case presented in Fig. 4.3 represents an approximate upper limit to the abundance at a given age

(it is not an exact upper limit, as the minor structural eﬀects of rotation described

119 above can cool the base of the convection zone somewhat; e.g. Eggenberger et al.

2012).

4.3.2. Inﬂation

Next, I assess the impact of radius inﬂation in the absence of rotational mixing. The left panel of Fig. 4.4 shows in colour the Li tracks of non-rotating 0.9M models ⊙ with a variety of mixing lengths, which induce the radius anomalies listed in the key.

The three stages of pre-MS depletion described above persist in inflated models, but the magnitude of depletion in the second stage is found to be highly sensitive to the stellar radius. Inflation reduces the temperature at the base of the convection zone, suppressing the rate of Li proton capture reactions and leading to a higher ZAMS abundance. The non-inflated, rotating tracks from Fig. 4.3 are shown in dotted grey for comparison. While increasing rotation enhances the destruction of Li, increasing inflation inhibits the destruction of Li. Both effects induce a dispersion, but the rotationally-induced spread lies below the standard prediction, and the inflationary spread lies above the standard prediction. The inflated models show no signs of additional depletion once on the MS, due to the lack of rotational mixing.

The right panel shows inﬂated isochrones at the age of the Pleiades, with the

0.9M locus denoted by the black dotted line. Li dispersions emerge generically at all ⊙ masses, as with rotation, but the magnitude of the dispersion imparted by inﬂation is a stronger function of temperature: the dispersion increases from ∼ 0.92 dex at

120 6000K to ∼ 1.30 dex at 5000K due to rotation, while increasing from ∼ 0.26 dex to

1.40 dex at the same temperatures due to inﬂation. Importantly, the dispersion at

fixed Teff is larger than the dispersion at fixed mass; radius inflation substantially reduces the Teff of stars, so the Teff of inflated, higher mass, Li-rich stars can be equal to that of non-inflated, lower mass, Li-poor stars. Ultimately, we see that a spread in radii of 10 per cent during the pre-MS can produce a Li dispersion that is small at temperatures typical of early G stars (∼ 6000K), and increases to nearly two orders of magnitude at temperatures typical of mid K dwarfs (∼ 4500K). These results are qualitatively similar to the findings of Somers & Pinsonneault (2014), who examined isochrones of constant mixing length instead of constant radius anomaly.

To summarise Section 4.3.1-4.3.2, increased rotation and increased inflation have the opposite effect for Li depletion: faster rotation leads to lower Li abundances, while higher inflation leads to greater Li abundances. Both effects can induce dispersions if they vary from star to star, but inflationary dispersions lie above standard predictions, and rotational dispersions lie below standard predictions.

However, both effects serve to reduce stellar Teff s below standard stellar model predictions, due to the increasing radius driven by rotational pressure support and reduced convective efficiency.

121 4.3.3. Rotation and inﬂation

I now present models which include both rotation and inflation. In this section, I permit any combination of these two parameters, for the purposes of exploring their combined impact on Li. Fig. 4.5 compares the tracks of rotating models with a 10 per cent radius anomaly (dashed black; slowly rotating in the left panel, rapidly rotating in the right) to three models from Section 4.3.1–4.3.2: a non-inflated, non-rotating model (solid grey), a model with a 10 per cent radius anomaly and no rotation (dashed grey), and a rapidly rotating model with no inflation (solid black; slowly rotating in left, rapidly rotating in right).

In the left panel, the inflated slow rotator track (dashed black) is nearly identical to the inflated non-rotator track (dashed grey), suggesting that the inflationary suppression of Li destruction dominates when rotation is mild. By contrast, the right panel shows that rapid rotation induces significant extra depletion relative to the non-rotating case in the presence of inflation. The non-rotating, inflated model

(dashed grey) is significantly more abundant than the rapidly rotating, inflated model (dashed black), which in turn is far more abundant than the rapidly rotating, non-inflated model (solid black). Interestingly, the final abundance of the rapidly rotating, inflated model (dashed black in right panel) is close to that of the slowly rotating, non-inflated model (solid black in the left panel). This demonstrates that the counteracting effects of rotational mixing and inflation can almost entirely cancel each other out with regard to Li destruction, allowing a rapidly rotating inflated star

122 to possess the same Li abundance as a slowly rotating non-inﬂated star at the age of the Pleiades.

To explore the mass dependence of this behaviour, I present isochrones of slow and rapid rotation at 125 Myr in the left and central panels of Fig. 4.6.

Pleiades data are also plotted as circles, and the dashed line signiﬁes the locus of

0.9M models. The slow rotating models make similar predictions to the inﬂated, ⊙ non-rotating models presented in Fig. 4.4, again demonstrating that the structural and hydrodynamic eﬀects of rotation are minimal at low angular velocity. The abundance spread resulting from a 10 per cent radius anomaly matches the Pleiades dispersion well, as did my non-rotating models in Somers & Pinsonneault (2014).

Compared to these models, the rapid rotating isochrones in the central panel are signiﬁcantly depleted. The 0 per cent anomaly isochrone is so heavily depleted by rotational mixing that it lies far below the data. However, the inﬂation of the 10 per cent anomaly isochrone suppresses rotational mixing enough to approximately trace the locus of rapid rotators below ∼ 5000K. This correspondence is driven in part by the suppression of Li depletion, and in part by the substantial reduction in the surface temperature of my models, demonstrated by the dotted line (0.9M locus). ⊙

Finally, the right panel of Fig. 4.6 reproduces the slow rotating, non-inﬂated isochrone from the left panel in purple, and the rapidly rotating, inﬂated isochrone from the central panel in peach. The result is intriguing: below ∼ 5700K, the rapid rotators appear more abundant in Li than the slow rotators. This is the

123 opposite sense from the pure rotation case, where faster rotation always leads to greater Li depletion, but the same sense as the pattern observed in the Pleiades.

Furthermore, the dispersion is small around 5500K, but increases substantially towards lower temperatures until reaching nearly 2 dex at 4500K, also reminiscent of the empirical Pleiades pattern. This behaviour suggests that a radius inﬂation which increases with rotation may be able to both produce the correct sense of the

Li–rotation correlation seen in the Pleiades, and accurately predict the width of the dispersion as a function of temperature. In order to assess this possibility, I consider in the following section the observational patterns resulting from diﬀerent functional dependencies of inﬂation on rotation.

4.4. Sleuthing the rotation–inﬂation connection

In Section 4.3.3, I argued that certain qualitative features of the Pleiades Li pattern are predicted if rapidly rotating stars are inﬂated relative to their slow rotating counterparts. In this section, I expand upon this idea by generating synthetic cluster distributions with a variety of inﬂation/rotation prescriptions, initialised to reproduce the Pleiades rotation distribution, and comparing them directly to the data.

To construct these synthetic distributions, I ﬁrst adopt a rule that explicitly links the degree of inﬂation to the rotation period at a benchmark age of 10 Myr

[δR = δR(Prot)]. Using this rule, I collapse my 3-dimensional model grid (mass, τD,

124 inﬂation) into two dimensions (mass, τD) by interpolating between models of ﬁxed

M and τD, but variable δR, to the location in parameter space where the inflation rule δR = δR(Prot) is satisfied. The result is a specific mapping between each choice of

M and τD (corresponding to a speciﬁc Prot at 10 Myr), and the relevant observables,

Teﬀ , A(Li), and Prot, at 125 Myr.

Next, I derive initial conditions by backwards modeling the empirical Pleiades pattern. For each Pleiad shown in Fig. 4.1, I determine the M and τD which maps onto the current day Teff and Prot, given a specific inflation law, and adopt the results as the starting model distribution. I then calculate the Li abundance at 125 Myr for the derived M and τD of each Pleiad, given the above rotation/inflation relationship.

This method is advantageous: since the synthetic cluster matches the temperature and rotation distribution of the present day Pleiades by construction, a comparison between the Li/rotation pattern of the synthetic cluster and the empirical pattern is unbiased by the sampling of the initial conditions. Finally, I randomly generate

Teﬀ and A(Li) errors from Gaussians with σ = 100K and 0.1 dex, respectively, and apply them to the models to simulate noise.

Using this methodology, I generate two limiting cases which represent distinct classes of physical models: no radius anomaly in any stars, regardless of rotation

(I); constant radius anomaly of 10 per cent in all stars, regardless of rotation (II). I further test a third limiting case, with radius anomalies in the range 0–10 per cent, uncorrelated with rotation (III). The methodology of Case III diﬀers somewhat from

125 Cases I and II, since no direct mapping exists between rotation and inflation in the uncorrelated scenario (see Section 4.4.3). With the synthetic distributions in hand, I search for three features of the empirical Pleiades pattern: the median abundance as a function of Teff , the width of the Li distribution as a function of Teff , and a positive correlation between Li abundance and rotation rate. In each case, the resulting distribution is found to be at odds with the empirical Pleiades pattern. Finally,

I demonstrate that a power law relationship between Prot and δR above a critical rotation period produces a qualitatively accurate synthetic distribution, and show that the agreement is insensitive to the details of the assumed mapping. I caution that the mass dependence, the normalisation, and indeed even the existence, of the correlation between rotation and inﬂation are at present unresolved (see Section

4.5.1). I therefore present this section as an exploratory exercise, and do not attempt a rigorous quantitative analysis.

4.4.1. Case I: no inﬂation

The ﬁrst limiting case assumes that no inﬂationary process operates on the pre-MS, and all stars can be described with a solar calibrated mixing length – hence δR(Prot)

≡ 0.0. This represents the ﬁducial prediction of stellar evolution theory.

The top left panel of Fig. 4.7 contains the Case I ‘map’, which shows the values of A(Li), Teﬀ , and Prot resulting from each M and τD grid point at the age of the

Pleiades. In this plot, lines connect models of equal mass but diﬀerent disc-locking

126 time. The size of the points denote the rotation rate, and the black dashed lines mark the approximate envelope of Pleiades data (see upper right panel). Line colours alternate to ease the distinguishing of mass tracks. Just as in the pure rotation models of Section 4.3.1, the stars with the shortest disc-locking times attain the most rapid rotation, and show the greatest degrees of Li depletion. The models bracket the lower envelope of the Pleiades distribution, but do not approach the upper envelope.

The synthetic distribution for this case, generated as described above, appears in the bottom left panel of this ﬁgure. As can be seen, the resulting Li pattern is inconsistent with empirical cluster distribution. The median abundance falls below the empirical median, the distribution width fails to increase substantially towards lower Teﬀ , and while the slowly rotating models approximately correspond to the locus of slow rotators in the Pleiades, the rapid rotators are too strongly depleted in Li to reproduce the observed pattern. As none of the aforementioned empirical features is reproduced by this limiting case, I strongly rule out the possibility that the observed Li pattern results purely from rotationally-induced mixing.

4.4.2. Case II: uniform inﬂation

The second limiting case assumes a ‘saturated’ inﬂation law; all stars possess the same radius anomaly regardless of rotation rate – hence δR(Prot) ≡ 0.1. This scenario is consistent with Jackson & Jeﬀries (2014a), who found that the average K dwarf

127 radius in the Pleiades is ∼10 per cent larger than expected from isochrones calibrated on old, inactive stars. Furthermore, uniform δR on the pre-MS is supported by

Preibisch et al. (2005) and Argiroffi et al. (2013), who found that nearly all stars in the pre-MS associations ONC and h Per are saturated, and might therefore have the same radius anomaly if magnetic field proxies correlate directly with radius inflation.

The upper middle panel of Fig. 4.7 shows the mapping for this scenario. The slowest rotating models tend to lie above and to the right of their counterparts in

Case I, as inﬂation suppresses Li destruction and reduces the surface temperature.

For the rapidly rotating stars, the decrease in Teﬀ is far more extreme, causing a strong shift to the right. The result is a substantial overlap of the mass tracks. This is not seen in the most massive tracks, since even the fastest rotators in these bins have spun down substantially by 125 Myr (see Fig. 4.3). The overall trend is a strong suppression of Li destruction in all stars, regardless of rotation.

The lower middle panel shows the resulting synthetic distribution. As in Case

I, the models occupy a band at fixed Teff that is far narrower than the empirical spread. Unlike Case I, the median abundance trend lies far above the empirical median, and traces the upper envelope of the Pleiades distribution. This signifies that a uniform suppression of depletion is too strong to reproduce the most Li-poor stars below ∼ 5500K. The location of the rapid rotating models approximately corresponds with the rapid rotators in the Pleiades, but they remain more depleted than slowly rotating models. Once again, the qualitative features of the Pleiades

128 pattern are not in general reproduced, ruling out uniform inﬂation from the epoch of pre-MS Li destruction to the present. I note however that Case I and Case II suggestively bracket the observed data range.

4.4.3. Case III: uncorrelated inﬂation

The ﬁnal limiting case assumes that no direct correlation exists between rotation and inﬂation. As stated above, a direct mapping such as those constructed for Cases

I and II cannot be defined for Case III, as rotation and inflation are by definition uncorrelated. Instead, I adopt the initial distribution of M and τD derived from the

Pleiades for Case I, randomly draw an inﬂation factor between 0–10 per cent for each individual star, and interpolate in the model grid to ﬁnd the 125 Myr distribution.

A physical scenario which may correspond to this limiting case is proto-stellar radius variations caused by accretion (Section 4.5.1).

The resulting synthetic distribution is presented in the bottom right of Fig.

4.7. Unlike the previous two cases, the model dispersion now matches the empirical dispersion reasonably well. In particular, it increases from ∼0.5 dex at the solar temperature to ∼1–1.5 dex below 5000K, in good agreement with the Pleiades. The median of the pattern appears to track the empirical median as well. However, the rotation distribution does not correspond to the observed pattern. Since rotation and inﬂation are uncorrelated, I see far too many slow rotators at the top of the distribution, and far too many rapid rotators at the bottom. For example, cool-ward

129 of 5000K, 21 out of the 32 stars with rotation periods greater than 2 days have A(Li)

> 2 in my synthetic distribution (65 per cent), compared to only 1 out of 15 in the empirical distribution. The probability of ﬁnding only 1 slow rotator out of 15 when

4 the expectation is 65 per cent is ∼ 5 × 10− , ruling out Case III at high conﬁdence.

This suggests that without some correspondence between rotation and inﬂation, the

Li–rotation correlation in the Pleiades cannot be produced.

4.4.4. Case IV: correlated inﬂation

None of the first three cases resemble the observed pattern, but they collectively point towards a resolution. As noted, the locus of slow rotators in the Teff –A(Li) plane is accurately predicted by Case I, the locus of rapid rotators is accurately predicted by Case II, and the Teff dependence of the median and width are produced by a range of radius anomalies from 0–10 per cent, as in Case III. If the Li depletion history of slow rotators was similar to those in the Case I mapping (i.e. no radius anomaly), and the depletion history of rapid rotators was similar to those in the

Case II mapping (i.e. ∼10 per cent radius anomaly), the Li–rotation correlation in the Pleiades would be produced, and the Teﬀ dependence of the median and dispersion may be preserved. Following this logic, my last class of models assumes a correlation between rotation rate and inﬂation factor, such that rapid rotators are physically larger than slow rotators.

130 I implement Case IV as follows. The maximum radius anomaly in my grid (10 per cent) is assigned to the fastest rotators, which achieve Prot ∼ 0.5 days at 10 Myr.

The radius anomalies of slower rotators are then determined by a power law, which extends down to a cut-oﬀ rotation period, below which stars are un-inﬂated. The functional form is given by Eq. 4.5.

β β β β 0.1 × P0% − P / P0% − P10% , P ≤ P0% δR =  ³ ´ ³ ´ (4.5)  0.0, P>P0%  

Here, P is the model rotation period at 10 Myr, P10% is the rotation period at which stars attain the maximum radius anomaly of 10% (set to 0.5 days as described above), and P0% is the rotation period below which radius inﬂation is absent. β represents the power-law exponent of the relation. Thus, stars rotating slower than

P0% at 10 Myr will have no radius inflation, stars rotating faster than P0% but slower than P10% at 10 Myr will have an inflation factor between 0 and 10 per cent, and stars spinning at P10% at 10 Myr will have a 10 per cent radius inflation. An example of the inflation law is shown in blue in panel A of Fig. 4.8 (see below for details).

While this formulation is clearly ad hoc, its features are motivated by data.

Rapidly rotating, magnetically active stars tend to show the greatest deviations from standard models (e.g. L´opez-Morales 2007), whereas slowly rotating, inactive stars generally agree with stellar models. This suggests that measurable inﬂation sets in above a critical rotation rate, as I have modeled. A power law form was motivated

131 by the power law dependence of chromospheric activity on rotation (e.g. Pizzolato et al. 2003), and my maximum anomaly was chosen to coincide with the upper envelope of claimed anomalies in the literature, as previously stated.

With this methodology, I ﬁrst test a linear scaling (β = 1), with a transition period P0% = 2 days. This version of Eq. 4.5 is shown in blue in panel A of Fig. 4.8, and the corresponding mapping between τD and δR is shown in red. Panel C shows the map for Case IV. The behaviour of the models is striking. Unlike Cases I and

II, where a shorter τD always results in a lower abundance, the response of Case IV models to shorter τDs is mass dependent. Higher mass models (e.g. 1.2M ) always ⊙ cool down and deplete more with greater rotation, whereas lower mass models (e.g.

0.8M ) show a more complicated relation. For long τD greater rotation induces ⊙ more depletion, since none of these models are inﬂated in our chosen mapping.

For intermediate τD greater rotation actually inhibits depletion, since intermediate rotation (τD ∼ 0.7 Myr) has a smaller effect on Li depletion than intermediate inflation (δR ∼ 0.05; see Fig. 4.4). Finally for short τD, mixing again dominates, and the Li abundance decreases with faster rotation. The overall trend is that shorter disc-locking times drive the models to the upper right of this figure, and increase their Pleiades-age rotation rate.

The Case IV synthetic distribution derived from the empirical Pleiades pattern appears in panel D. Unlike the ﬁrst three cases, several salient qualitative features of the empirical distribution are reproduced by this mapping. First, the locus of

132 slow rotators roughly corresponds to the lower envelope of the Pleiades distribution, just as in Case I. Second, the most rapid rotators lie above and to the cool side of this locus, roughly corresponding to the locus of rapid rotators in the empirical pattern, as in Case II. This morphology is due in part to the decreased rate of

Li destruction on the pre-MS, and in part to the reduced Teff of rapidly rotating, inflated stars. Third, a significant dispersion sets in around 5500K and increases towards cooler temperatures, in excellent agreement with the Pleiades. It can be understood why this occurs at 5500K from the Case IV map. Rapid rotation displaces stars horizontally towards cooler temperatures in the Teff –A(Li) plane; this horizontal path coincides with the slow rotators above 5500K, but lies above the slow rotators at lower temperatures, since these stars are strongly depleted. The

Teff decrement due to rotation and inflation is of a similar magnitude for all masses, but the effect on the observed distribution is more apparent in the sub-solar regime.

One location where the models conspicuously fail is the coolest rapid rotators; four stars near 4500K are predicted to have A(Li) > 1.5 by the models, but are actually more heavily depleted by 0.5-1 dex. This may reﬂect a deﬁciency in the modeling of radius anomalies, which does not account for changes as a function of mass and evolutionary state. For instance, the true maximal radius anomaly could decrease with mass, so that depletion factors among the fastest spinning K dwarfs are larger than I have predicted.

133 Despite the ad hoc nature of the δR(Prot) scaling, this model succeeds at accurately predicting several puzzling features of the observed Pleiades distribution.

However, one could reasonably contend that ﬁne tuning may be required to achieve this qualitative agreement. To address this concern, I assess the generality of the

Case IV agreement by calculating synthetic cluster distributions resulting from different choices of β and P0% in Eq. 4.5 (Fig. 4.9). From left to right I increase the power law exponent (β), and from top to bottom I decrease the rotation rate at which inflation sets in (P0%). The initial conditions were derived for each individual case using the specified inflation prescription, and the same run of Teff and A(Li) errors was applied to each. From this plot, I see that some features of the Case IV mapping are generic, such as the population of rapid rotators to the upper right of the main locus of slow rotators, and the onset of dispersion at ∼ 5500K. Other features of the synthetic pattern vary from mapping to mapping, such as the number of rapid rotators that lie near the bottom of the abundance distribution below 5500K

(approximately half for P0% = 1 day, none for P0% = 4 days; the dependence on β is negligible). Never the less, since the important qualitative features of the Pleiades pattern are generically reproduced by each considered version of Eq. 4.5, I conclude that ﬁne tuning is not essential for the conclusions of this section.

To summarise Section 4.4.4, the three qualitative features of the empirical pattern described at the beginning of Section 4.4 are reproduced by a relationship between rapid rotation and inﬂation, and these qualitative features are largely

134 insensitive to the exact choice of mapping function. I conclude that Case IV strongly outperforms each of the limiting cases, and produces an observational pattern consistent with the Pleiades.

4.5. Discussion

I have presented the first consistent physical model that replicates the qualitative pattern of Li and rotation observed in the Pleiades. Section 4.4.4 delineated the properties of a successful solution, so I now turn to plausible explanations for the rotation–inflation correlation that I have inferred. I consider both the inhibition of convection by magnetic fields, and the impact of mass accretion during the proto-stellar phase. Next, I discuss a method for detecting radius dispersions on the pre-MS and beyond, as a test of the scenario and as a means to constrain the putative connection between rotation and inflation.

4.5.1. The mechanism of inﬂation

Our paradigm assumes the existence of a mechanism which reduces the temperature at the base of the convection zone across the observed range of rapid rotators on the pre-MS. This behaviour naturally results from an increase in the stellar radius, so I have developed the models in the context of radius anomalies correlating with rotation. There is observational support for this eﬀect from young clusters and associations (Littlefair et al. 2011; Cottaar et al. 2014), and from pre-MS eclipsing

135 binaries (Stassun, Mathieu & Valenti 2007; Stassun et al. 2014). In this section, I discuss two possible mechanisms for inducing a rotation-radius correlation.

I have modelled radius inﬂation as a product of inhibited convective eﬃciency.

This may result as a direct consequence of rapid rotation, when extreme Coriolis forces begin to govern the behaviour of convective particles, but I focus on the suppression of convection produced by rotationally-generated magnetic ﬁelds.

Magnetic fields in the envelopes of low mass stars can disrupt the motions of charged particles when they attempt to cross field lines. This leads to an increase in the convective energy flux per unit area in the non-magnetic regions, forcing the radius to expand in order to match the flux emanating from the core (Spruit

1982a; Andronov & Pinsonneault 2004; Chabrier, Gallardo & Baraﬀe 2007; Feiden

& Chaboyer 2013, 2014). An important consequence of this effect is the formation of star spots on the surface, which result from the inefficient heat flux induced by magnetism. Several authors have taken this approach, and considered the impact of spots on the radii of stars (Chabrier, Gallardo & Baraffe 2007; Jackson, Jeffries

& Maxted 2009; Jackson & Jeﬀries 2013, 2014a; Somers & Pinsonneault 2015b).

Polytropic models show that a filling factor of ∼0.35–0.51 is sufficient to achieve a K dwarf inflation ∼10 per cent, consistent with filling factors ∼0.2–0.4 found on active

G and K dwarfs (O’Neal et al. 2004). This behaviour appears particularly effective in stars with deep convection zones (Jackson, Jeffries & Maxted 2009), and so may be effective on the pre-MS. Magnetic fields can also provide pressure support in

136 stellar interiors, thus working to counteract the force of gravity similar to rotational support (Mullan & MacDonald 2001; MacDonald & Mullan 2009, 2012, 2013). Each of these eﬀects serves to puﬀ the radius up in stellar models, leading to a reduced temperature in the convection zone.

While these eﬀects would naturally produce the type of correlation I require, activity studies of pre-MS clusters have found that most young stars of ages ∼ 1–15

Myr are saturated, meaning that the strength of their magnetic ﬁeld proxies (i.e.

X-rays) no longer correlate with rotation rate (Preibisch et al. 2005; Argiroﬃ et al.

2013). The predominate interpretation of this result is that magnetic ﬁelds cease to increase in strength beyond the critical Rossby number at which saturation sets in (RN = Prot/τCZ ∼ 0.1; Pizzolato et al. 2003). If true, it would appear diﬃcult for convection to be progressively inhibited by rapid rotation among pre-MS stars.

However, an alternative interpretation is that magnetic field proxies decouple from the strength of the surface field at the onset of saturation, so that magnetic fields can continue to grow with rotation without impacting proxies. For instance, saturation could represent a threshold above which chromospheric and coronal heating cannot increase, but the spot filling factor can (e.g. O’dell et al. 1995). Alternatively, the spot filling factor could saturate, but the spot temperature could continue to decrease with greater rotation. Several studies have attempted to measure directly the magnetic field strengths of saturated stars and have found mixed results (e.g.

Saar 1996, 2001; Reiners, Basri & Browning 2009), but their eﬀorts are hampered

137 by the extreme diﬃculty of inferring Zeeman broadening from stars whose lines are already highly broadened by rotation (Reiners 2012). Due to these observational issues, the continued inhibition of convection with greater rotation in the saturated regime remains unresolved at present, but further work in this area will ultimately favour or disfavour this mechanism as a viable avenue.

Mass accretion may also play an important role in determining pre-MS radii.

Proto-stars accrete the majority of their mass during the ﬁrst ∼ 105 yr, when the

3 initial core of M ∼ 10− M gains material from the surrounding over-density in its ⊙

8 7 natal molecular cloud. Accretion likely progresses at a modest level (10− –10− M ⊙

1 6 4 1 yr− ), interspersed with short bursts of vigorous accretion (10− –10− M yr− ) ⊙ until the ﬁnal mass is assembled (see Baraﬀe, Vorobyov & Chabrier 2012). A major unknown factor in this picture is the amount of thermal energy deposited during these intense accretion bursts. ‘Cold accretion’ scenarios assert that all of the accreted thermal energy is radiated away at the photosphere, so only mass

(and not entropy) is added to the proto-star (Hartmann, Cassen & Kenyon 1997, and references therein). This additional mass leads to a contraction of the radius.

Alternatively, ‘hot accretion’ scenarios suggest that a substantial fraction of the thermal energy of the accreted material is absorbed into the stellar envelope. This type of accretion can increase stellar radii (e.g. Palla & Stahler 1992).

In either case, a dispersion in radii will arise from diﬀering accretion histories, and may persist throughout the pre-MS due to the lengthy thermal readjustment

138 times. This leads to a range of Li depletion eﬃciencies during the epoch of standard burning (Baraﬀe & Chabrier 2010). If hot accretion is accompanied by substantial angular momentum deposition, or if the rotation period of cold accretors remains

fixed during radius contraction due to magnetic interactions with circum-stellar material, then the necessary rotation correlation will be likewise present. This scenario makes two interesting predictions. One, that the rotational history of stars is intrinsically linked to their accretion history. Two, that radius inflation is only a transient event on the pre-MS, and therefore will be gone by the age of the Pleiades, leaving behind a Li dispersion as a fossil record of early time inflation. The debate in the literature about hot versus cold accretion scenarios remains unsettled (Baraffe et al. 2009; Hosokawa, Offner & Krumholz 2011; Baraffe, Vorobyov & Chabrier 2012), so further work remains to determine precisely how accretion impacts stellar radii on the early pre-MS.

While these mechanisms are fundamentally different, their impact on the structure at the base of the convection zone will be analogous. Any increase in the stellar radius moves the base of the convection zone outwards, reducing the rate of Li depletion, and any decrease in the stellar radius has the opposite effect. I therefore believe that the results of Section 4.3-4.4 hold in the presence of a radius dispersion, regardless of the underlying cause. Given recent efforts to probe the radius–rotation connection in young stars, constraints may soon be placed on the connection between these two properties during the pre-MS.

139 4.5.2. Testing the inﬂation paradigm

Jackson & Jeﬀries (2014a) claimed that the average radius of the Pleiades K dwarfs is inﬂated by ∼10 per cent, but they did not attempt to characterise the distribution about this average. If a radius dispersion was present in the Pleiades at ∼10 Myr, as required by the theory, then a remnant of the dispersion might still exist today.

Detection of a radius dispersion at 125 Myr would have important implications not only for the evolution of Li, but for the structure of stars on the pre-MS, the behaviour of saturated stars, and the eﬀects of spots and magnetic ﬁelds on stellar radii.

A viable avenue for testing whether a spread is present in the Pleiades is through obtaining surface gravities for a large number of cluster members. If rapidly rotating Pleiads are physically larger than slowly rotating Pleiads, a dispersion in log g at fixed colour arises due to three compounding effects. First, a radius increase of ∼10 per cent naturally reduces the surface gravity (g ∝ M/R2) compared to non-inflated stars at fixed colour. Second, the most inflated stars are very rapidly rotating; rapid rotation reduces Teff at fixed mass (Fig. 4.4), enhancing the effect at

fixed temperature. Third, since inflation also reduces Teff , the most inflated stars at fixed colour are also the most massive. Surface gravity decreases with increasing mass, because R is proportional to M in the solar regime (g ∝ M/R2 ∝ M/M2 ∝

1/M), leading to an additional reduction in the log g of the most inﬂated (most massive) stars at ﬁxed colour.

140 For a 10 per cent dispersion in radius, the models predict that the log g dispersion could be as large as 0.2 dex at ﬁxed colour. This signal may be detectable through granulation-induced light curve ﬂicker (Bastien et al. 2013). The characteristic time-scale of the formation and dissolution of convective cells on the surface of stars is proportional to the surface gravity. This induces a correlation between the power of short time-scale luminosity variations and the strength of surface gravity, permitting the inference of stellar log gs with an accuracy of 0.1–0.15 dex (e.g. Bastien et al. 2013, 2014). This signal can be extracted from high precision, long baseline, short cadence photometry, such as that obtained by the Kepler satellite. Although its mission has ended, Kepler’s descendant K2 will observe a

field containing the Pleiades in early 2015 (Howell et al. 2014), and despite the degraded photometric quality, may be able to detect granulation flicker at the necessary level. This offers a unique opportunity to detect and characterise the distribution of surface gravities, and search for a dispersion about the inflated mean.

This test will open the door for using Li abundances to understand differential pre-MS stellar structure, but much work remains to be done in order to correctly interpret the information. First, we must determine the mechanism driving the putative connection between rotation and inflation. If magnetic fields are responsible, how do they impact stellar structure in the saturated regime? If accretion is responsible, is the cold or hot scenario more appropriate, and how do they impart the angular momentum distribution? If differential radius inflation

141 on the pre-MS is not responsible for the observed Li dispersion, then how is it formed? Second, we must understand the mass, rotation, and age dependence of radius inflation on the MS. K2 will place constraints on this mechanism, since it will measure surface gravities in a large number of clusters of different ages, and track the evolution of anomalies along the MS. Third, the systematic effects plaguing the precise determination of Li abundances and effective temperatures in rapidly rotating

K dwarfs must be resolved. These include the shifting spectral energy distributions driven by surface spots, the blending of the Li absorption line with nearby lines due to rotation, and the effects of active chromospheres on line formation (see King et al. 2010, and references therein). Once accurate Teff s and abundances can be determined, the true rate of Li depletion for individual stars on the pre-MS can be derived, and the magnitude of the Li dispersion at fixed mass, and hence the degree of radius inflation on the pre-MS, will be revealed. Finally, observations of the surface gravity distribution of the Pleiades can only provide circumstantial evidence in favour of the theory. Detecting a dispersion in radii correlated with rotation in a

∼10 Myr old cluster, or detecting a correlation between rotation at Li abundance at ﬁxed temperature in a 10–20 Myr cluster, will serve as the decisive test of an inﬂationary origin for the observed Li–rotation correlation.

142 4.6. Summary

In this chapter, I have explored the simultaneous impact of rotation and radius inﬂation on the depletion of lithium from stellar models during the pre-MS. Our goal is to describe the generic features of a mechanism which would bring the predictions of rotational mixing into concordance with the well-known Li–rotation correlation in the K dwarfs of the Pleiades. Such a mechanism must suppress Li destruction in rapid rotators relative to slow rotators, thus overcoming the trend predicted by rotational mixing calculations. As radius anomalies ∼10 per cent are sometimes found in rapidly rotating stars, and a increased radius is known to inhibit

Li depletion in standard models, I was motivated to test whether a correlation between rotation and radius inﬂation on the pre-MS (induced by some physical process neglected in standard stellar models) could be responsible for the observed pattern.

I first reviewed the impact of rotation and inflation on the evolution of Li abundances. In accordance with previous studies, I find that the depletion of

Li is a strong function of both properties, such that rotation increases the rate of Li destruction, and radius inflation, modeled by a reduction of the mixing length parameter, decreases the rate of Li destruction. While this behaviour is straight forward, the picture becomes far more complicated when the two effects are combined. Rapid rotation and substantial radius inflation (10 per cent) enhances Li depletion for stars with M > M , but inhibits Li depletion for stars with M < M , ⊙ ⊙ 143 relative to non-rotating benchmarks. This is the first new result of the work, and implies that if the radii of just the rapid rotators in the Pleiades were inflated by

10 per cent during the pre-MS, while the radii of the slow rotators agreed with standard predictions, then a positive correlation between the rate of rotation and the abundance of Li at 125 Myr could arise in K dwarfs, in agreement with observations.

Next, I assessed whether this qualitative notion could explain the observed relationship between Li and rotation in the Pleiades. To do this, I generated synthetic cluster distributions derived from the empirical Teﬀ /rotation distribution of the Pleiades which obey various inﬂation/rotation prescriptions, and compared them to the empirical Pleiades pattern. First, I found that three limiting cases, which represent distinct classes of physical models, were unable to reproduce the

Pleiades pattern: no radius inflation in all stars regardless of rotation, uniform radius inflation of 10 per cent in all stars regardless of rotation, and radius inflation uncorrelated with rotation. Next, I adopted a prescription where rapidly rotating stars are more inflated than slowly rotating stars, and found that observed pattern in the Pleiades is generically produced. The dispersion in Li remains small around the solar temperature and increases to greater than an order-of-magnitude towards cooler temperatures, precisely as is found in the Pleiades. Moreover, the median abundance trend is accurately reproduced, and rapidly rotating stars are found to be more rich in Li than slowly rotating stars in the K dwarf regime. Furthermore,

I found that these qualitative features are insensitive to the precise details of the

144 rotation/inflation prescription I adopt as long as more rapid rotation leads to greater inflation, demonstrating that fine tuning is not required to achieve this level of agreement. This is the first consistent physical explanation for the Pleiades

Li pattern to date, so I conclude that it is a strong candidate for explaining the observed pattern.

After establishing that a connection between rotation and inflation could explain the Pleiades pattern, I discussed what sort of physical mechanisms might lead to this correlation. The inhibition of convection due to strong magnetic fields in rapid rotators could lead to such a correlation if field strengths continue to grow with rotation in the saturated regime. Accretion could also alter the radii and rotational histories of proto-stars, plausibly inducing the observed correlation.

Finally, I suggest that the remnants of a pre-MS radius dispersion may still be present in the Pleiades, and a survey of stellar surfaces gravities in this cluster could place strong constraints on the radius dispersion needed at 10 Myr to produce the observed pattern. The ongoing K2 mission oﬀers a valuable opportunity to perform this experiment.

145 3.5 Initial A(Li)

3.0

2.5

2.0 A(Li) 1.5

1.0 Prot < 1 day 1 day < P < 3 days 0.5 rot 3 days < Prot < 6 days

6 days < Prot 0.0 6500 6000 5500 5000 4500 Effective Temperature (K)

Fig. 4.1.— Rotation and lithium in the Pleiades. The Li abundance pattern of the Pleiades (see Section 4.2.3). The sizes of point represent stellar rotation periods, as indicated by the key, and clearly illustrate the enhanced Li abundance of rapid rotators redward of 5500K. The black line represents a standard stellar model prediction for the Li pattern of this cluster, taken from Somers & Pinsonneault (2014), and the dotted black line shows the initial Li abundance of these models.

146 2.0 δR =

0.00 1.8 )

ML 0.02 α 1.6 0.04

1.4 0.06

0.08 Mixing Length ( Length Mixing 1.2 0.10

1.0 1.2 1.1 1.0 0.9 0.8 0.7 Mass (M ⊙)

Fig. 4.2.— Radius anomaly versus mixing length. The mixing length parameters αML required to produce a radius anomaly δR at 10 Myr, as a function of mass (see Section 4.2.1.

147 Age (Myr) Effective Temperature (K) 0.1 1 10 100 6000 5500 5000 4500 0.1 0.1 Fast 0.9M ⊙ 1.2M ⊙−0.7M ⊙ Rotation Period (days) Teff ∼5250K at 125 Myr Fast at 125 Myr

1 1

Slow Rotation Period (days) Period Rotation Slow 10 10 3.5 3.5

3.0 3.0

2.5 Slow Slow 2.5

2.0 2.0 A(Li) 1.5 1.5 τD = 0.1 Myr

A(Li) τ = 0.2 Myr 1.0 D Fast 1.0 τD = 0.4 Myr

0.5 τD = 1.0 Myr 0.5

τD = 3.0 Myr 0.0 Fast 0.0 τD = 6.0 Myr No Rotation -0.5 -0.5 0.1 1 10 100 6000 5500 5000 4500 Age (Myr) Effective Temperature (K)

Fig. 4.3.— Influence of rotation on lithium depletion. The impact of rotation on Li depletion in non-inflated models. Left: The evolution of the surface rotation period (top) and the envelope Li abundance (bottom) in 0.9M non-inflated models. A ⊙ range of initial disc-locking times produces a range of Li depletion factors at the age of the Pleiades, which mainly set in between ∼5–20 Myr. Faster rotation leads to greater depletion in these models. Right: Isochrones of rotation and Li at the age of the Pleiades. A large range of rotation rates persist at this age, given the initial rotation distribution. A Li dispersion forms at all temperatures in the models, and fans out below the standard prediction (black dashed line). A black dotted line shows the location of the 0.9M models represented in the left column. ⊙

148 3.5 3.5

δR = 0.1 δR = 0.1 3.0 Rotation Inflation 3.0

2.5 2.5 δ = 0.0 2.0 R 2.0 A(Li) δ = 0.0 1.5 R 1.5 δR = 0.00 A(Li) δ = 0.02 1.0 R 1.0 δR = 0.04 0.5 0.5 δR = 0.06 δ = 0.08 0.0 R 1.2M ⊙−0.7M ⊙ 0.0 0.9M δR = 0.10 ⊙ at 125 Myr -0.5 -0.5 0.1 1 10 100 6000 5500 5000 4500 Age (Myr) Effective Temperature (K)

Fig. 4.4.— Influence of radius inflation on lithium depletion. The impact of inflation factors of 2-10 per cent on Li depletion in non-rotating models. Left: Li tracks of 0.9M models. A greater radius anomaly decreases the rate of depletion during the ⊙ 5–20 Myr age range, imparting a dispersion at 125 Myr. This dispersion lies above the standard prediction, opposite of the rotationally-induced dispersion (grey dotted lines). Right: Inflated isochrones at the age of the Pleiades. A large dispersion is present which increases towards lower temperatures. The black line in this figure is identical to the black dashed line in Fig. 4.3.

149 3.5 3.5

3.0 3.0

2.5 2.5

2.0 2.0 A(Li)

A(Li) 1.5 1.5

1.0 No Rotation, No Inflation No Rotation, No Inflation 1.0 No Rotation, Max Inflation No Rotation, Max Inflation 0.5 Slow Rotation, No Inflation Fast Rotation, No Inflation 0.5 Slow Rotation, Max Inflation Fast Rotation, Max Inflation 0.0 0.0 1 10 100 1 10 100 Age (Myr) Age (Myr)

Fig. 4.5.— Influence of rotation and inflation on lithium tracks. Left: The simultaneous effects of inflation and slow rotation. A standard model (solid grey), an inflated non-rotating model (dashed grey) and a non-inflated, slowly rotating model (solid black) are compared to an inflated, slowly rotating model (dashed black). Inflation dominates the evolution of Li when rotation is negligible. Right: Same as left panel, except with rapid rotation instead of slow rotation. The depletion effects of maximal inflation (10 per cent) and maximal rotation (τD= 0.1 Myr) nearly cancel each other out at the age of the Pleiades, as the inflated rapid rotator (dashed black) reaches a similar abundance as the standard model (solid grey) at 125 Myr. It is clear that both inflation and rotation can strongly influence Li depletion.

150 3.5 3.5 Slow Rotating Rapid Rotating Models Models Rapid Rotator 3.0 3.0 δR = 0.0 δR = 0.1 2.5 2.5 δR = 0.1

2.0 2.0 A(Li) δR = 0.0 Slow Rotator

A(Li) δ = 0.0 1.5 R 1.5 δR = 0.0 δR = 0.10 δ = 0.08 1.0 R 1.0 δR = 0.06 Prot < 1 day δR = 0.04 1 day < P < 3 days 0.5 rot 0.5 δR = 0.02 3 days < Prot < 6 days δ = 0.00 R 6 days < Prot 0.0 0.0 6000 5500 5000 4500 6000 5500 5000 4500 6000 5500 5000 4500 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 4.6.— Influence of rotation and inflation on lithium depletion patterns. Pleiades- age isochrones for models with both inflation and rotation. Left: Slow rotating models with a variety of inflation factors. The Pleiades data from Fig. 4.1 are also plotted. The Li dispersion resulting from a 10 per cent anomaly in slow rotators matches the observed dispersion as a function of Teff well. Centre: Same as left panel, but with rapid rotation. The most Li-rich, rapidly rotating Pleiads coincide with the maximally inflated, rapidly rotating isochrones. Right: The slow rotating, non-inflated isochrone from the left panel and the rapidly rotating, maximally inflated isochrone from the central panel are reproduced, along with lines matching models of equal mass. Below 5500K, these models succeed at producing the inverted Li–rotation pattern, as seen in the Pleiades.

151 3.5 3.5 3.5 1.20 1.10 1.00 0.90 3.0 3.0 0.80 3.0

2.5 2.5 2.5 1.20 1.10 2.0 1.00 2.0 2.0

A(Li) A(Li) A(Li) P = 0.25 day 1.5 0.90 1.5 1.5 rot Prot = 0.5 day

1.0 1.0 1.0 Prot = 1.0 day 0.70 Prot = 2.0 day P = 8.0 day 0.5 Case I Map 0.5 Case II Map 0.5 rot 0.80 δ (P ) = 0.0 δ (P ) = 0.1 Empirical Pattern 0.0 R rot 0.0 R rot 0.0 6500 6000 5500 5000 4500 4000 6500 6000 5500 5000 4500 4000 6500 6000 5500 5000 4500 4000 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

3.5 3.5 3.5

3.0 3.0 3.0

2.5 2.5 2.5

2.0 2.0 2.0

A(Li) 1.5 A(Li) 1.5 A(Li) 1.5

1.0 1.0 1.0

0.5 Case I SD 0.5 Case II SD 0.5 Case III SD δ (P ) = 0.0 δ (P ) = 0.1 δ & P uncorrelated 0.0 R rot 0.0 R rot 0.0 R rot 6500 6000 5500 5000 4500 4000 6500 6000 5500 5000 4500 4000 6500 6000 5500 5000 4500 4000 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 4.7.— Testing limiting rotation/inflation connections. Maps and synthetic distributions for the three limiting cases described in Section 4.4.1–4.4.3. The lines in the maps are equal mass, and the points represent different disc-locking times, whereas each point in the synthetic distribution represents a forward modeled Pleiad. The size of each point reflects the rotation rate at 125 Myr, as indicated by the key in the upper right panel. Case I assumes that all stars, regardless of rotation, have solar calibrated mixing lengths. Case II assumes that all stars possess 10 per cent radius anomalies, as might be expected from uniform saturation. Case III assumes no correlation between rotation and inflation. Each of these cases fails to reproduce the observed pattern, shown in the upper right panel and represented by approximate envelopes in each of the figures.

152 Prot at 10 Myr (days) 0.1 1 10 0.10 A 3.5 B 3.0

R 0.05 δ 2.5

0.00 2.0 0.10 A(Li) 1.5 1.0

R 0.05 δ 0.5 Empirical Pattern 0.00 0.0 0.1 1 10 6500 6000 5500 5000 4500 4000 τ (Myr) D Effective Temperature (K)

3.5 C 3.5 D

1.10 1.00 3.0 1.20 3.0 2.5 2.5 0.90 2.0 2.0 0.80

A(Li) 1.5 A(Li) 1.5 1.0 1.0

0.5 0.70 0.5 Case IV Map 0.0 0.0 Case IV SD 6500 6000 5500 5000 4500 4000 6500 6000 5500 5000 4500 4000 Effective Temperature (K) Effective Temperature (K)

Fig. 4.8.— Testing the best-case rotation/inflation correlation. The results of assuming a linear scaling between rotation and inflation (β = 1, P0% = 2 days in Eq. 4.5). A: A graphical representation of the inflation function. The top panel shows the mapping between rotation at 10 Myr and the radius anomaly, and the bottom shows how this maps on to the distribution of disc-locking times. B: The empirical Pleiades pattern reproduced. C: The map resulting from this inflation paradigm. The effect of greater rotation in both mass dependent, and non-uniform as τD is decreased. The general trend is to push rapidly rotating stars to the upper right portion of this panel, near the empirical upper envelope shown in dashed black. D: The synthetic distribution resulting from this mapping. This paradigm reproduces the empirical median, the empirical distribution as a function of Teff , and the observed Li–rotation correlation.

153 β = 0.5 β = 1.0 β = 2.0 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K) 6000 5500 5000 4500 6000 5500 5000 4500 6000 5500 5000 4500

P0% = 3 3

2 2 A(Li) 1 day 10 10 10

A(Li) 8 8 8 1 6 6 6 1 4 4 4 RA (%) RA (%) RA (%) RA 2 2 2 0 0 0 0 0.5 1 2 4 8 0.5 1 2 4 8 0.5 1 2 4 8 0 Prot at 10 Myr Prot at 10 Myr Prot at 10 Myr

3 3

2 2 A(Li) 2 day 10 10

A(Li) 8 8 1 6 6 1 4 4 RA (%) RA (%) RA 2 2 0 0 0 0.5 1 2 4 8 Empirical Pattern 0.5 1 2 4 8 0 Prot at 10 Myr Prot at 10 Myr

3 3

2 2 A(Li) 4 day 10 10 10

A(Li) 8 8 8 1 6 6 6 1 4 4 4 RA (%) RA (%) RA (%) RA 2 2 2 0 0 0 0 0.5 1 2 4 8 0.5 1 2 4 8 0.5 1 2 4 8 0 Prot at 10 Myr Prot at 10 Myr Prot at 10 Myr

6000 5500 5000 4500 6000 5500 5000 4500 6000 5500 5000 4500 Effective Temperature (K) Effective Temperature (K) Effective Temperature (K)

Fig. 4.9.— Varying the strength of the rotation/inflation correlation. Synthetic distributions resulting from a variety of choices of β and P0%, along with the individual mappings between rotation and inflation illustrated in inset panels. The qualitative features of the pattern shown in Fig. 4.8 are a general feature of power law inflation models, but the exact distribution is sensitive to the details of the equation and the underlying rotation distribution. The value of P0% has a stronger effect on the distribution than β, since the fraction of stars in the rapidly rotating locus depends on the absolute number of highly inflated members.

154 Chapter 5: The Impact of Starspots on Pre-Main-Sequence Stellar Evolution

5.1. Background

Precision measurements of fundamental stellar parameters active, late-type stars are systematically larger by ∼5-15% than predictions from modern evolution codes, a phenomenon I refer to as radius inflation. This effect has been claimed in open clusters (e.g. Jackson & Jeffries 2014a) and in field stars (e.g. Popper 1997; Torres &

Ribas 2002; Ribas 2003; L´opez-Morales & Ribas 2005; Morales et al. 2008; Boyajian et al. 2012); on the main sequence (MS), and the pre-MS (e.g. Stassun, Mathieu

& Valenti 2007); and for stars above and below the fully-convective boundary

(e.g. Torres, Anderson, & Giménez 2010; Feiden & Chaboyer 2012a). The cause remains controversial, but a vital clue comes from the claimed correlation of radius inflation with magnetic activity (Torres et al. 2006; López-Morales 2007; Morales et al. 2008; Stassun et al. 2012). Mechanisms proposed to induce this correlation include the magnetic inhibition of convection (e.g. Chabrier, Gallardo & Baraffe

2007), magnetic ﬁelds adding internal pressure support (e.g. Mullan & MacDonald

2001), and starspots (e.g Torres et al. 2006), though each are likely at work to some

155 degree in active stars. Although evidence is accruing that such radius anomalies are real, interpreters must be cautious as complications such as mis-estimated opacity

(e.g. Terrien et al. 2012), unaccounted systematics in interferometic measurements

(Casagrande et al. 2014), and the impact of spots on radius measurements in eclipsing binaries (Morales et al. 2010) may be mistaken for signatures of radius inﬂation.

In Chapter 3, we found that radius inflation during the pre-MS has a strong impact lithium abundances of young stars. Radius inflation reduces the temperature at the base of the surface convection zone during the pre-MS, leading to a suppressed rate of Li destruction in standard stellar models. If stars of equal mass and age have different intrinsic sizes during the pre-MS, they will destroy different quantities of Li during the pre-MS, thus providing a mechanism for inducing the

Li abundance dispersion observed in the K dwarfs of the Pleiades and other young clusters (e.g. Soderblom et al. 1993a). In Chapter 4, we further showed that radius inﬂation can suppress Li destruction caused by rotational mixing on the pre-MS

(e.g. Pinsonneault et al. 1990). This implies that if rotation correlates with an increased radius on the pre-MS, as claimed in some works (e.g. Littlefair et al.

2011; Cottaar et al. 2014), the Li-rotation correlation observed in the Pleiades could also be produced. This is important because rotational mixing is correlated with rotation rate, producing a trend of opposite sign relative to the data. Both of these experiments were constructed as general investigations of the eﬀect of radius inﬂation

156 on Li abundances, conducted by varying the mixing length parameter (e.g. Chabrier,

Gallardo & Baraﬀe 2007), and as such did not model a unique physical mechanism.

In this chapter, I expand upon this work by exploring a specific potential cause of radius inflation: starspots. Spots have long been argued to affect the luminosities, temperatures and radii of stars (e.g. Spruit 1982a; Spruit & Weiss 1986), but in only a few cases have their effect on the pre-MS been examined (e.g. Jackson & Jeffries

2014a). Following Spruit & Weiss (1986), I implement a treatment of starspots into my evolutionary code, and explore the impact of spottiness on pre-MS and ZAMS stellar parameters, the inference of masses and ages from the Hertzsprung-Russell

(HR) diagram, the observed colors of stars with non-zero spot temperatures, and the destruction of lithium. By comparing my model predictions to data, a consistent picture emerges: pre-MS stars have a range of radii at ﬁxed mass and age, correlated with rotation, which persists at the ZAMS.

The text is organized as follows. I discuss my evolutionary models in Section

5.2, with a particular emphasis on my treatment of spots (Section 5.2.2). With my models calculated, I ﬁrst consider the impact of spots during the pre-MS and at the

ZAMS on stellar radii, luminosities, and Teff s in the mass range 0.1-1.2M (Section ⊙ 5.3). Next, I discuss how spots might introduce errors into the quantification of masses and ages using isochrones (Section 5.4.1), and how spots affect observed colors (Section 5.4.2). Then, I examine how spots impact the destruction of lithium, both in the context of age determinations with the lithium depletion boundary

157 technique, and in the context of Li abundance spreads observed in young main sequence clusters (Section 5.5). In Section 5.6, I synthesize the results of Section

5.3−Section 5.5, compare my quantitative ﬁndings to other work in the literature, and discuss some outstanding questions for starspot modeling. Finally, I summarize my conclusions in Section 5.7.

5.2. Methods

5.2.1. Standard stellar models

I calculate non-rotating, solar-metallicity stellar models with the Yale Rotating

Evolution Code (YREC; e.g. van Saders & Pinsonneault 2013). For the purposes of calibration, I assume the Sun is unaffected by the presence of spots; this is reasonable, given the Sun’s minimal spot coverage. I adopt a present day solar heavy element abundance of Z/X = 0.02292 from Grevesse & Sauval (1998), and choose the hydrogen mass fraction (X) and the mixing length coefficient (αML) that reproduce the solar luminosity and radius for a 1M model at 4.568 Gyr. The final solar ⊙ calibrated values are X = 0.71172, Z = 0.018923, and α = 1.92531. My models ⊙ use the 2006 OPAL equation of state (Rogers, Swenson & Iglesias 1996; Rogers &

Nayfonov 2002), model atmospheres from Allard et al. (1997), high temperature opacities from the opacity project (Mendoza et al. 2007), low temperature opacities from Ferguson et al. (2005), and nuclear cross-sections from Adelberger et al. (2011).

I treat electron screening using the method of Salpeter (1954).

158 My models are initialized above the Hayashi track, and evolved to the age at which 99% of the initial deuterium abundance has been destroyed (e.g. Stahler 1988).

I deﬁne this time as t = 0 for each individual model. This approach represents a physical scenario: as proto-stars contract and build up mass through accretion, their interiors increase in temperature. Eventually, the core becomes hot enough to fuse deuterium, a process which partially supports the star against gravitational contraction; this is the so-called “deuterium birth line”. Proto-stars travel along this sequence towards higher mass as accretion continues to supply fresh deuterium fuel to the central engine. Once the accretion rate subsides, the star will begin to contract towards the hydrogen burning main sequence. My zero point provides a consistent criteria for the starting age of all models, and ensures that models of equal mass but diﬀering spot properties are in equivalent evolutionary states when compared at early ages.

5.2.2. Spots

Implementation

My spot methodology, based on the treatment by Spruit & Weiss (1986), incorporates two eﬀects: the inhomogeneous photosphere in the presence of spots, and the inhibition of convection in magnetic regions. To simulate this behavior,

I divide my stellar models into two zones: un-spotted and spotted. I account for the ﬂux redistribution by explicitly altering the radiative transport gradient in the

159 surface convection zones of my models, and the inhomogeneous surface by properly treating the outer boundary condition (see below).

The spots in my models are controlled by two parameters: the filling factor fspot, defined as the ratio of the spotted surface to the total surface area, and the spot temperature contrast ratio xspot, defined as the temperature of the spotted surface divided by the temperature of the un-spotted photosphere. The surface-weighted average of the two zone temperatures defines the temperature of the model at a given radius.

4 4 4 T = (1 − fspot)Tamb + fspotTspot, (5.1)

Tspot is the area-averaged temperature of the penumbral and umbral spot regions, and Tamb is the temperature of the ambient, un-spotted region. By substituting

Tspot = xspotTamb, I ﬁnd an expression for the total luminosity,

2 4 4 L = 4πR σSBTamb(1 − fspot + fspotxspot), (5.2)

and the ﬂux in the ambient regions,

4 Famb = Ftot/(1 − fspot + fspotxspot). (5.3)

This formulation implies a modeling degeneracy between fspot and xspot, since the ﬂux redistribution is actually controlled by the redistribution parameter,

4 αspot = (1 − fspot + fspotxspot). (5.4)

160 αspot can be thought of as one minus the eﬀective blocking area of totally black

(xspot = 0) spots, a parameter typically called β (= 1 − αspot). Eq. 5.3 governs the ﬂux enhancement in the non-magnetic regions of the model. I implement this enhancement into my evolution code by dividing the radiative gradient in convective regions by αspot,

∇rad,amb = ∇rad/αspot. (5.5)

Since fspot and xspot fall in the range 0–1 by definition, αspot is always less than or equal to one, meaning that spots can only increase the radiative flux in the un-spotted regions. The luminosity and effective temperature of the star can be found by solving Eqs. 5.1 and 5.2 using the values of Tamb, Tspot, and xspot at the surface where the optical depth τ = 2/3. To model the changing temperature of spotted regions with depth, I hold the difference in temperature between the two zones fixed (Tamb − Tspot = const.). The result is that spots are only significant near the surface, where T is low. I test the importance of spot depth on my models in

Section 5.6.3, and ﬁnd that assumptions about the subsurface morphology impact radius, luminosity, and Teﬀ predictions by ∼ 1 % at most.

My spot treatment involves a crucial implicit assumption: the spotted and un-spotted atmospheric regions are in pressure equilibrium. This means that if the spotted regions contain a magnetic ﬁeld which produces a pressure Pspot,mag, the gas

161 1 pressure in the two regions will be such that Pamb,gas = Pspot,gas + Pspot,mag . This requirement permits us to obtain the surface pressure in the un-spotted region, and apply it equally in both zones. Psurf is obtained in my evolution code by using a model atmosphere, which returns a pressure Psurf for a given Teﬀ and log g. For consistency, I call my atmosphere routine using the temperature of the un-spotted, ambient surface:

Psurf = Psurf (Tamb,surf , log g). (5.6)

My approach differs from previous studies which have considered the effects of magnetic activity on stellar properties. Spruit & Weiss (1986) used a similar spot formulation to investigate the impact on stellar parameters, but focused primarily on the ZAMS, and did not consider the effect of non-zero spot temperatures on the location of stars in the color-magnitude diagram. In a recent series of papers, Feiden

& Chaboyer (2012b) examined the consequences of magnetic fields in stellar interiors by directly incorporating a 1-dimensional approximation of the induction equation into their model physics (Feiden & Chaboyer 2013, 2014). This approach differs from ours by considering a one-zone, homogeneous surface boundary condition, compared to the two-zone, inhomogeneous approach I have adopted. However, since magnetic fields are the underlying cause of surface spots, my work compliments theirs by approaching the same mechanism from a different perspective. In addition,

1 A further assumption of magnetic and gas equiparition therefore implies Pamb,gas = 2×Pspot,gas, a fact I make use of in Section 5.6.2

162 Jackson & Jeﬀries (2014a,b) have recently examined the impact of surface spots on the pre-MS radii of late-type stars, and the lithium destruction rates of stars with

M < 0.5M . Their procedure relied on calibrating polytropic models on published ⊙ evolutionary tracks, and then considering the impact of flux-blocking spots at the surface. By contrast, I employ a self consistent evolutionary code to produce my calculations, and thus anticipate somewhat different results (Section 5.6.2). Each of these studies differed in their treatment from my present study, and I consider my results complimentary to theirs.

Adopted spot properties

In this section, I select a range of spot properties to investigate in my models. I am interested in the range of stellar properties that could be produced by spots, as well as their dependence on evolutionary state and mass. Most measured ﬁlling factors to date have been for dwarfs, sub-giants, and giants (see tab. 9 in Berdyugina 2005).

While very few pre-MS stars have similar measurements, perhaps the most direct comparison are the spotted, tidally-synchronized sub-giants known as RS CVn stars.

RS CVns are structurally similar to pre-MS stars in the outer regions, as they have deep convective envelopes and low gravities relative to the MS, and thus host similar environments for spot emergence. Claimed ﬁlling factor among this class reach 40 to 50%, such as in the cases of HU Vir, UX Ari, and HR1099 (O’Neal et al. 1998,

2001), demonstrating that deep envelopes paired with moderate (5 − 10 d) rotation periods can produce a high level of spot coverage. The T Tauri star V410 Tau has

163 a measured surface spot coverage of 32-41% (Petrov et al. 1994), supporting the notion that pre-MS stars can be similarly spotted. Furthermore, Zeeman-Doppler imaging has conﬁrmed average surface ﬁelds of up to a few kG on pre-MS stars (e.g.

Reiners 2012), comparable to the surface ﬁelds of highly active, heavily spotted main sequence stars (see below). Collectively, these arguments support the hypothesis that pre-MS stars can host spots which approach 50% coverage.

Similar ﬁlling factors have been claimed for young main sequence stars. For example, the rapidly rotating (Prot ∼ 1 to 4 d) dwarfs LQ Hya, V833 Tau, and EQ

Ver host ﬁlling factors of 40 to 50 % (O’Neal et al. 2001, 2004). It may appear surprising at ﬁrst that the slower rotating RS CVn stars (e.g. Prot ∼ 10.4 d for

HU Vir; O’Neal et al. 1998) can be as spotted as the faster rotating ZAMS stars.

However, the convective overturn timescale of sub-giants is much longer than main sequence dwarfs, implying a smaller Rossby number (ratio of rotation rate to overturn timescale) at ﬁxed rotation period. Magnetic properties appear to scale more closely with Rossby number than with absolute rotation (e.g. Noyes, Weiss &

Vaughan 1984; Pizzolato et al. 2003), so it is reasonable to expect equivalent spot properties at slower rotation rates for sub-giants. This argument applies equally well to pre-MS stars, whose average rotation rates are slower at 3 Myr than at the ZAMS

(e.g. Littlefair et al. 2010), but whose convection zones are deeper.

I conclude that fspot = 0.5 is a reasonable assumption for the upper limit of pre-MS and ZAMS ﬁlling factors, and I adopt this. I further adopt xspot = 0.8,

164 characteristic of the range of observed spot temperature ratios (0.65 to 0.85; e.g.

Berdyugina 2005). A fuller treatment would include a mapping of spot temperature and ﬁlling factor as a function of age, metallicity, mass, and rotation rate; the current exercise is aimed at a diﬀerent purpose, namely constraining the potential impact of star spots on observables. I do note that the Pleiades lithium dispersion

(see Section 5.5) does provide evidence for a range of spot properties in pre-MS stars, as we argued in Somers & Pinsonneault (2015a).

5.3. Eﬀect on stellar radii

5.3.1. The pre-main sequence

I ﬁrst explore the impact of spots on stellar model radii during the pre-MS.

Evolutionary tracks representing both fully convective stars(M < 0.35M ) and stars ∼ ⊙ which develop radiative cores (0.35M < M < 1.2M ) are shown in the upper panel ⊙∼ ∼ ⊙ of frame A in Fig. 5.1. Solid black lines represent the radius as a function of time for un-spotted models, and the dashed black lines represent the same for models half covered in 80% temperature contrast spots (fspot = 0.5, xspot = 0.8, αspot ∼ 0.7), referred to hereafter as my base case. The diﬀerence in radius between equal mass, un-spotted and spotted models as a function of age is reported in the lower panel.

Spots lead to an inﬂated radius at all ages, but the magnitude evolves during the pre-MS. For each mass, the radius enhancement gradually increases with

165 age, initially peaking around 8-12%, and ﬁnally stabilizing on the MS. The age dependence of pre-MS radius inﬂation results primarily from a timing argument: the rate of pre-MS contraction is proportional to stellar luminosity, and because un-spotted stars are more luminous than their spotted counterparts on the pre-MS

(see below), they will have contracted further at a given age. This explains the gradual increase in ∆R with time, and leads to a conspicuous transient spike in ∆R at the onset of main sequence burning, as the un-spotted star has reached the ZAMS while the spotted star remains on the late pre-MS. However the internal structure is also altered in a mass dependent fashion: changes to the eﬃciency of convection impact higher mass models (e.g. Andronov & Pinsonneault 2004), while changes to the surface boundary condition are important at lower mass (e.g. Spruit & Weiss

1986).

Spotted models show corresponding anomalies in both Teff and luminosity. The average Teff for my base case is lower by up to 7% for all masses, and corresponds to a zero point shift in the Hayashi track. Pre-MS luminosity anomalies therefore evolve according to L/R2 ∼ const. Luminosities of the spotted models are lower by up to 18% and 10% during the early and late pre-MS for high mass stars, and lower by up to 25% and 10% during the early and late pre-MS for low mass stars. I explore consequences of these Teff and L anomalies in Section 5.4 and Section 5.5.

166 5.3.2. The zero-age main sequence

Once at the ZAMS, radius anomalies converge to a stable, mass-dependent pattern.

The black line in the upper panel of frame B in Fig. 5.1 shows my base case radius anomalies at the age when 0.1% of the core hydrogen has been burned. As can be seen, fully convective stars have a weakly mass-dependent anomaly of ∼4%, and higher mass stars generally have larger inﬂation factors with local maxima at 0.5M and 1.2M , and a local minimum at 0.7M . Anomalies resulting from ⊙ ⊙ ⊙ diﬀerent values of fspot are also shown. Unsurprisingly, the radius enhancement increases self-similarly with greater spot coverage, and the change in ∆R with fspot is approximately linear. These results are generally consistent with Spruit & Weiss

(1986), though the second minimum around 0.8M is not present in their ZAMS ⊙ models. This may be due to diﬀerences in the adopted standard model physics.

This radius enhancement in spotted models is predominately a result of the altered boundary conditions at the surface. The photospheric pressure is higher for the hotter ambient photosphere than it would be for a model with Tsurf = Teﬀ at ﬁxed radius. The model expands as a result in order to reach equilibrium

(P ∝ M 2/R4 in homology), which lies at higher radius and lower surface pressure.

This reduced P leads to a corresponding decrease in P throughout the surface convective zone, due to its adiabatic stratiﬁcation. For fully convective stars, the pressure reduction reaches the nuclear burning core where it directly suppresses the rate of reactions, and thus lowers the luminosity. The blue line in the lower panel

167 of frame B shows ∆L as a function of mass, revealing a significant decrease in the luminosity of fully convective stars due to my base case spots. This effect is less pronounced in higher mass stars, whose convection zones have receded from the core and therefore exert less influence on the central engine (Kippenhahn & Weigert

1990). The decreasing luminosity anomaly from 0.3–0.6M leads to a spike in the ⊙ radius and temperature (red line in frame B) anomalies, which peak around 0.5M ⊙ and 0.7M respectively. Note that above ∼0.5M , ∆R ≈ −2∆T , in good agreement ⊙ ⊙ with observational studies (e.g. Morales et al. 2008).

The peculiar structure of the dependence of ∆R on mass can be understood by considering the spotted and un-spotted mass-radius relationships plotted in the upper panel of frame C in Fig. 5.1. As can be seen, between 0.1 and 1.0M the ⊙ slope of the mass-radius relation at first decreases, then increases, decreases, and increases again. These “wiggles” result from mass-dependent changes in stellar structure and energy generation, such as the onset of a radiative core, the decreasing mass of the convective envelope, and the burgeoning dominance of CNO hydrogen processing. The lower panel shows both mass-radius relations detrended with the line M = R (in solar units). The inclusion of spots alters the masses at which these transitions occur, shifting the onset of slope changes to lower mass, and thus leading to significant structure in the mass dependence of radius inflation.

168 5.3.3. Comparison with data

In this section, I compare the mass-radius relationship predicted by my models to literature data. The left panel of Fig. 5.2 displays my predicted relationships for un-spotted (black line) and spotted (blue line) models at the ZAMS, and for un-spotted models at 10 Gyr (dashed red). These are compared to M ≤ 1.0M ⊙ eclipsing binary data from Torres, Anderson, & Gim´enez (2010) and Feiden &

Chaboyer (2012a) (red and black, respectively). The binaries generally form a tight sequence bracketing the spotted isochrone below 0.8M , and fan out into a ⊙ larger range above 0.8M , attesting the importance of age eﬀects at a solar mass ⊙ and above. In the right panel, the data falling within the dotted parallelogram are detrended with the standard ZAMS isochrone, and compared to the spotted and 10

Gyr isochrones. The data occupy an envelope ∼ 4 − 7% at the low mass end, and increase to ∼12% at 0.7M , demonstrating the inflated radius problem discussed in ⊙ Section 5.1. Two-thirds of the detrended stars fall within the shaded blue region, and all but one fall within the combined effects of age and spots2 (dotted purple line). This shows that the radius anomalies resulting from realistic spot parameters in my models are consistent with empirical constraints on radius inflation.

I ﬁnd no clear evidence for higher average anomalies around 0.5M compared ⊙ to 0.75M (see Section 5.3.2), but a host of systematic issues forbid any present ⊙

2Old stars are normally not expected to be heavily spotted, but short period eclipsing binaries are often tidally locked, permitting rapid rotation at late ages.

169 conclusions. For example, stars of equal mass but unequal metal abundance will have diﬀering radii, but I ﬁnd that this dispersion is only ∼ 2% for [Fe/H] ∈ [−0.3, +0.3].

Additionally, surface spots can increase the inferred transit time for eclipsing binaries, thus leading to systematic over-estimates of up to ∼3% in the radius

(e.g. Morales et al. 2010). Finally, age effects will contribute to inferred anomalies, as suggested by Fig. 5.2. Feiden & Chaboyer (2012a) have recently argued that accounting for these effects reduces most radius anomalies to around 5%, instead of the often quoted range of 5-15%, in good agreement with my expectations. In any case, it appears that spot-induced radius effects are of a similar magnitude as the well-known radius anomalies in eclipsing binaries, and should not be neglected when interpreting these observations.

5.4. Eﬀect on inferred masses, ages, and colors

Measuring the masses and ages of pre-MS stars is a challenging endeavor, given the rapid evolutionary timescales and a host of observational issues (e.g. Hartmann 2001), but is of great importance for studies of the initial mass function, circum-stellar discs, the early angular momentum evolution of stars, and the formation of planets.

The ideal approach would be to compare calibrated pre-MS isochrones with the HR diagram of a cluster, and to read oﬀ the masses and ages of its members. In reality, this procedure is complicated by a variety of observational and theoretical issues, which broadly fall into three categories.

170 Observational errors. In young clusters, extinction is highly variable from source to source, binarity status is often unknown, and stellar magnitudes may vary on short and long timescales due to rotation, disc obscuration, and accretion. These eﬀects form a base level of uncertainty.

Theoretical issues. Pre-MS evolutionary tracks lack a good calibration data set, and diﬀerent suites make varying predictions for the birth line location, the bolometric corrections, and the rate of Hayashi contraction. This uncertainty severely limits one’s ability to derive masses and ages in young clusters.

Non-uniform parameters. Even if these issues were resolved, the possibility remains that stellar parameters such as luminosity and radius vary between cluster members at ﬁxed mass and age, perhaps due to physics associated with rotation, magnetic activity, or accretion.

The picture is further complicated by intra-cluster age spreads, which mimic the appearance of non-uniform stellar parameters by producing luminosity dispersions at fixed Teff . In order to separate these effects, it is important to asses the impact that non-uniform parameters would have on coeval stars.

In Section 5.3, I showed that spots induce shifts to the radii, luminosities and

Teﬀ s of stars during the pre-MS, and can therefore generate non-uniform parameters among young cluster members. In this section, I explore the impact of these shifts

171 on the Hertzprung-Russell and color-magnitude diagrams of young clusters, and discuss their implications for understanding pre-MS stars.

5.4.1. Spot eﬀects in the Hertzsprung-Russell diagram

I ﬁrst constrain myself to the theoretical HR diagram, and consider the impact of spots on the temperature and luminosity of my stellar models. This for now ignores the impediments to obtaining these parameters brought on by spots (see Section

5.4.2), and assumes the true surface Teﬀ and L are known.

The red dashed lines in the left panel of Fig. 5.3 show un-spotted tracks for a range of masses, run from the deuterium birth line (dotted red line) to 50

Myr. The blue dashed lines represent the same for my base case spotted models.

Isochrones of 3 Myr are over-plotted in solid red and blue, with masses periodically enumerated. The results of Section 5.3 are reflected in this figure: spots reduce the intrinsic luminosity and Teff of stars in a mass-dependent fashion. The fractional

Teff displacement ranges from ∼6–8% from the highest to the lowest mass, which corresponds to a nearly mass-independent decrement of 270 K. The fractional luminosity reduction ranges from ∼10–22% from high to low mass, corresponding to shifts of 0.1 – 0.01 L . A range of spot filling factors will therefore induce a ⊙ significant dispersion in L at fixed Teff . However, even in my extreme base case the dispersion does not reach the order-of-magnitude spread observed in some young clusters (e.g. Hillenbrand 1997). Thus spots may be an important contributor to

172 young cluster dispersions, but not their sole cause. I emphasize that these shifts are not an observational bias, but a change in the true temperature and luminosity of pre-MS stars.

Pre-MS Hayashi evolution proceeds at approximately fixed Teff , with L declining with age. My models demonstrate this behavior, but also reveal an important result: spots alter the Teff at which this evolution occurs. Spots displace stars downward and to the right in the plane of the HR diagram, and consequently spotted stars appear younger and less massive when interpreted with standard isochrones. To quantify these errors, I use the un-spotted evolutionary tracks of Fig. 5.3 to infer the masses and ages of a range base case spotted models, and derive a set of correction factors to the resulting parameters. The results appear in the right panel of Fig. 5.3.

These factors operate in the following sense: if the mass and age of a 50% spotted star are derived from its HR diagram location using un-spotted tracks, the true mass and age will be given by the derived values multiplied by their respective correction factors (see the caption for a speciﬁc example). Factors for a variety of masses, ages, and ﬁlling factors can be found in Table 5.1.

Mass correction factors (solid lines) for solar-type stars are as large as 1.3×, but decrease with inferred age since spotted and un-spotted evolutionary tracks begin to overlap near the horizontal Henyey track (e.g. the 0.9M tracks). Mass errors ⊙ increase substantially towards the fully convective regime, where spotted star masses can be under-estimated by up to 2.2×, and never less than 1.7×, regardless of age.

173 The larger errors for low mass stars result from the stronger impact of spots on fully convective luminosities, and the temporal stability of the errors results from the long contraction timescale of low mass stars. Age correction factors (dashed lines) are more severe. For solar-type stars, ages are under-estimated by up to 2.5×, and become more accurate as stars get older. At 0.2M , ages will be under-estimated ⊙ by up to a factor of 10, with the largest errors coming at the earliest epochs. This order-of-magnitude shift results from the large fractional errors incurred when considering values near zero – for instance, an under-estimate of 1 Myr for a 1.1

Myr old star produces a correction factor of 11. They also relate to the decreasing luminosity of the deuterium birth line with mass: spotted stars are born at a higher luminosity at fixed Teff , and thus lie above un-spotted stars at fixed age (Fig. 5.3).

Consequently, spotted stars interpreted with un-spotted tracks will appear far younger than their true ages, thus requiring a correction factor much larger than unity. I note that the magnitude of the age corrections depends somewhat on how one deﬁnes t = 0, but that mass correction factors are largely insensitive to this choice.

I note an interesting prediction of these models. In my physical scenario, proto-stars travel up the deuterium birth-line during the assembly phase, slowly gaining mass through accretion. They are prevented from substantial contraction by deuterium burning feedback, and prevented from persisting in a higher luminosity state by the short Kelvin-Helmholtz time-scales (e.g. Baraﬀe et al. 2009). The region

174 of the HR diagram above the standard deuterium birth line therefore represents a

“forbidden region”, in which stars with low spot coverage may only reside for a short time. By contrast, heavily spotted stars naturally occur in this location due to their intrinsically lower average surface temperatures. For example, the spotted 3 Myr isochrone in Fig. 5.3 lies entirely above the un-spotted birth line below ∼ 0.5M . ⊙ Such stars persist in the forbidden region for several tens of Myr, given their slow contraction rates, and should therefore dominate the population coolward of the standard deuterium birth line. This feature provides a direct observational probe of the prevalence of spotted stars in young clusters.

5.4.2. Spot eﬀects in the color-magnitude diagram

In Section 5.4.1, I drew Teﬀ and luminosity directly from my models to predict

HR diagram shifts. While valid if Teff and L are known, this approach neglects an important fact: spotted stars are inhomogeneously hot across their surface. This will have implications for the spectrum of emission from the photosphere, and may introduce color anomalies relative to pristine stars of equal Teff . To investigate this issue, I transform my models into color space using a two-zone model, where the emission is the area-weighted composite of the SEDs from the surfaces of temperature Tamb and Tspot, whose values are derived from the model Teff using

Eq. 5.1. This transformation is accomplished by using the bolometric corrections of An et al. (2007). Colors are rarely used to interpret pre-MS stars as numerous

175 complicating factors (circum-stellar discs, heavy extinction, etc.) render them un-reliable. I therefore advance my models to 125 Myr for comparison with data in the Pleiades, but note that analogous eﬀects occur on the pre-MS.

The top left of Fig. 5.4 shows the results for MV vs. V − K. The red line represents un-spotted models from 0.5-1.0M , and the dark blue line shows my ⊙ spotted models, transformed using the two-zone approach (xspot = 0.8, fspot = 0.5).

The spotted models appear redward of the standard models, though the shift is not as extreme as in the theoretical HR diagram (shown in the bottom right). This is because the un-spotted surface temperature is higher than the model Teff , thus enhancing the short wavelength emission and producing bluer colors. These models are compared to Pleiades data from Kamai et al. (2014), supplemented by rotation data from Hartman et al. (2010). Pleiads rotating faster than a two day period are shown as open circles, and slower stars are shown as black dots. The slow rotators generally agree with my un-spotted models, but the fastest rotators are typically more red by ∼0.1-0.2 magnitudes. As my spotted models also appear more red in this color band, this supports the interpretation that spots are affecting the color of the most active cluster members (e.g. Stauffer et al. 2003; Jackson & Jeffries 2014a), though I under-predict the magnitude of the shift. The models of Jackson & Jeffries

(2014a) found that spots induce a blue-ward shift in this plane, in contrast to the red-ward shift I ﬁnd. This discrepancy may be related to diﬀering choices of the spot temperature contrast (see below).

176 Next, I present isochrones in two additional color coordinates: MV vs. V − IC , and MV vs. B − V . As can be seen, the spotted models appear bluer in shorter wavelength colors. The spotted isochrone lies nearly on top of the un-spotted isochrone in V − IC , and completely blue-ward of it in B − V , similar to the behavior of the rapidly rotating data. This is due to the progressively stronger sampling of the

Wein tail as one considers shorter wavelengths. It has long been known that the K dwarfs of the Pleiades are blue in B − V , and red in V − K, compared to isochrones calibrated on older stars. Stauffer et al. (2003) attributed this behavior to spots and plages brought on by magnetic activity, which respectively enhance the long and short wavelength emission. My models suggest that qualitatively similar behavior is possible without plages, as the extra blue light can result from a slight increase in the temperature of the un-spotted photosphere. In reality, there are several effects I have neglected in my models which impact the detailed location of CMD isochrones, such as plages, faculae, and other activity-related phenomena. A more complete and detailed model is therefore necessary to make definitive quantitative statements.

These two-zone models were calculated using the same combination of fspot and xspot. As discussed in Section 5.2.2, these parameters are degenerate from a modeling perspective, since cooler spots with a lower surface area can redistribute the same amount of ﬂux as warmer spots with a higher surface area. This degeneracy is graphically represented in the bottom right of Fig. 5.5; the curved lines denote surfaces of constant ﬂux redistribution (my base case implies αspot ∼ 0.7). However

177 this modeling degeneracy does not require that the parameters be observationally degenerate, as the spectrum of photospheric emission depends on the temperatures and relative fractional coverage of the two zones.

I therefore present CMD isochrones calculated with diﬀerent degenerate combinations of xspot and fspot in Fig. 5.5. Each panel shows my un-spotted isochrones in red, and isochrones derived through the two-zone approach for four degenerate choices of xspot and fspot (see the colored dots in the bottom right panel).

In each case, we see that lower spot temperatures (lower xspot) give rise to bluer emission, since the decreasing flux of the spots leads to increasing dominance of the global emission by the hot ambient surface. This effect saturates at sufficiently low xspot and asymptotes to totally black spots. The saturation value is a strong function of the color bands: for example, xspot = 0.6 models are nearly indistinguishable from black spots in B − V , while they remain separated in V − IC and V − K. In the limiting case of black spots, the entire flux budget originates from the ambient regions, leading to bluer emission in all bands. This behavior is consistent with the

ﬁndings of Spruit & Weiss (1986), who considered the eﬀect on colors of black spots

(see their ﬁg. 2), but did not explore the inﬂuence of temperature contrast.

The fact that the observed emission of spotted stars is a composite of two zones may hold crucial implications for the spectra of young stars. Ionization fractions will be non-uniform across the surface, the strength of Zeeman broadening and splitting will diﬀer wildly from location to location, and the cool spotted regions may permit

178 molecular species which are too fragile for the hot photosphere. Given these eﬀects, it is clear that the observed spectrum will be diﬀerent than a pristine star of equivalent

Teﬀ , which may complicate the inference of spectroscopic stellar parameters and metal content. Furthermore, these isochrones show that multi-band photometry holds great value as a tool for breaking the degeneracy between fspot and xspot, which limits the conclusions of many observational studies of starspots. For example, a given spot modulation signal may increase due to greater (non-axisymmetric) spot coverage, or due to cooler spots. Combining spot modulation with isochrone comparisons may help to constrain the relative contribution of spot area and spot temperature. While I have exhibited the importance of this eﬀect, a full exploration of requires additional important stellar physics as described above, and is therefore outside the scope of this discussion.

5.5. Eﬀect on lithium depletion and cluster dating

Lithium has long been used as an indicator of youth in late-type stars. Li is destroyed

6 at TLi ∼ 2.5 × 10 K for typical stellar densities, and consequently depletes from the surface of stars when the base of the convective envelope reaches this temperature on the pre-MS. For stars more massive than ∼0.5M , the convection zone retreats ⊙ and cools below TLi near the end of the pre-MS, thus freezing out the surface Li abundance and imprinting a characteristic mass-dependent pattern at the ZAMS.

For stars between 0.5M and the sub-stellar boundary, Li in the outer layers will be ⊙

179 completely destroyed before reaching the MS. Li is also destroyed for some objects below the sub-stellar boundary (0.08M

In this section, I explore the impact of spots on Li destruction in two mass regimes: fully convective stars, which have been widely employed to measure cluster ages, and FGK dwarfs, whose depletion properties have long been at odds with standard theoretical predictions.

Perhaps the most important utility of Li is the measurement of young cluster ages through the lithium depletion boundary (LDB) technique (Basri et al. 1996;

Bildsten et al. 1997). When the central temperature of a fully convective star reaches

TLi on the pre-MS, Li is destroyed throughout the star in just a few Myr. The rate of Hayashi contraction, and thus the precise age at which Li destruction occurs, depends strongly on the masses of individual stars. As a consequence, there exists at a given age a critical mass below which stars retain their primordial abundance, and above which Li has been annihilated. Identiﬁcation of this transition mass, and the subsequent inference of cluster age by comparison with stellar models, comprises the

LDB technique. This method has been favored by many authors, due to its relative insensitivity to errors in the physical inputs of stellar evolution codes (∼ 3 − 8%

Burke et al. 2004). However, Jackson & Jeﬀries (2014b) recently showed that

180 because spots reduce the rate of pre-MS contraction (similar to my ﬁndings), the age at which TLi is reached in the interior of fully convective stars increases, introducing a bias into LDB ages inferred from spotted stars. In the following, I perform a similar analysis with my evolution code, and compare the results in Section 5.6.2.

To quantify this effect, I present models of M < 0.5M in the left panel of Fig. ⊙ 5.6. The lines in this figure represents the age at which 95% of the primordial Li abundance is destroyed as a function of bolometric luminosity in un-spotted (black line/squares) and spotted (magenta lines/squares) models. The fractional delay in the onset of Li destruction is greater than the fractional luminosity reduction, implying that derived ages will be under-estimates. The right panel of Fig. 5.6 shows the fractional age error incurred by using un-spotted evolutionary models to measure the lithium depletion boundary from stars with three different spot

ﬁlling factors, as a function of the true age. The measurement precision remains roughly constant from ∼15 to 60 Myr, and decreases sharply thereafter, due to the progressively stronger L reduction in lower mass stars (Section 5.3.1). For my maximal spot coverage, the fractional age error begins around 13% at 15 Myr, peaks at ∼ 15% at 30 Myr, and decreases below 5% after 100 Myr.

These numbers indicate errors for a uniformly spotted population. If spot properties vary from star to star, the eﬀect will be to eliminate the one-to-one mapping between mass and Li destruction age, thus blurring out the otherwise sharp depletion boundary. The strength of activity may also depend on spectral

181 type (Mohanty & Basri 2003; Schmidt et al. 2015), particularly near the sub-stellar boundary (SpT ∼ M8) in which case different values of fspot should be considered for different masses of the depletion boundary. LDB ages have long been in conflict with ages derived from isochrones, in the sense that LDB ages are systematically higher than main sequence isochrone fitting, and the spot-induced errors serve to exacerbate this conflict. I note however that the two techniques can be brought into better agreement by the inclusion of convective overshooting on the upper main sequence (e.g. Soderblom et al. 2014).

Next, I evaluate the impact of spots on Li destruction in higher mass stars.

As discussed above, stars which develop radiative cores cease surface Li depletion when the convective envelope retreats and cools near the end of the pre-MS. This occurs at earlier ages for higher mass stars, so the amount of Li preserved at the

ZAMS increases with effective temperature. Standard theoretical models anticipate a strong mass trend with little scatter at the ZAMS (Pinsonneault 1997). Contrary to this picture, large (> 10×) Li dispersions at fixed Teff have been identified in the

K-dwarfs of several young clusters (e.g. Soderblom et al. 1993a; Balachandran et al.

2011). Such dispersions further correlate with rotation, so that the fastest spinning stars tend to be the most Li-rich. We argued in Somers & Pinsonneault (2014) and

Somers & Pinsonneault (2015a) that if some rotation-correlated mechanism inﬂated stars relative to standard predictions on the pre-MS, then a rotation-correlated Li dispersion would naturally emerge in open clusters around 10-15 Myr. A dispersion

182 in Li at the ZAMS thus represents a fossil record of non-uniform stellar parameters among pre-MS stars during the epoch of Li burning. I have now implemented a specific mechanism which predicts radius inflation on the pre-MS (Section 5.3.1), and which is known to correlate with rotation in some stars, so I examine its effect on Li depletion in K-dwarfs.

Fig. 5.7 shows the Li abundances of 0.6–1.2M stellar models at 125 Myr (the ⊙ age of the Pleiades). Each line represents a diﬀerent fspot, given by the key in Fig.

5.6 (xspot = 0.8 in each), and each dotted line connects models of equal mass. As anticipated, different filling factors lead to different Li abundances at the ZAMS, with the dispersion surpassing a factor of ten at some masses. I have also plotted Teff ,

A(Li), and rotation information for a sample of stars in the Pleiades, assembled in

Somers & Pinsonneault (2015a) from data in Soderblom et al. (1993a) and Hartman et al. (2010). Point size indicates rotation period, from which the Li-rotation correlation can be readily identiﬁed below ∼ 5000 K. As can be seen, the abundance dispersion in these data roughly corresponds to models in the range fspot= 0.0–0.45

(αspot = 1.0 − 0.73). This analysis suggests that if rotation correlates with spottiness on the pre-MS as it does on the MS, spot-induced radius inﬂation naturally produces the Li-rotation correlation in this cluster (see Somers & Pinsonneault (2015a) for a lengthy discussion). This in turn implies an early (∼ 3 − 15 Myr) origin for the

Pleiades Li dispersion, a hypothesis which may be tested by obtaining Li abundances for clusters of age ∼10-15 Myr. Finally, while many stars are more depleted than the

183 un-spotted models above 5000 K, this discrepancy could be explained by rotational mixing (Somers & Pinsonneault 2015a), errors in the input physics of the models

(Somers & Pinsonneault 2014), or could indicate that the Pleiades was born with a sub-solar initial Li abundance (A(Li) < 3.31).

5.6. Discussion

I have implemented a simple spot model into my stellar evolution code, and assessed its eﬀect on stars during the pre-MS and on the ZAMS. I have particularly focused on the impact of spots on, 1) stellar structure as a function of mass; 2) the isochrone-derived masses and ages of pre-MS stars; 3) The colors of spotted stars near the ZAMS; 4) the depletion of lithium from stellar envelopes. In this section, I synthesize my results and consider some implications, compare my results with other works who consider magnetic models, and discuss some outstanding issues in both the physical interpretation and modeling of spotted stars.

5.6.1. A radius dispersion during the pre-main sequence

I have shown that if a range of spot properties (i.e. surface magnetic ﬁelds) exist at

ﬁxed mass and age during the pre-main sequence, the result is non-uniform stellar parameters among the members of young open clusters. But does this scenario accord with observational evidence?

184 In the case of lithium, I find a conspicuous spread in abundances at fixed temperature among the G- and K-dwarfs of open clusters shortly after the age of pre-MS Li burning. This spread cannot be explained by atmospheric effects (e.g.

King et al. 2010), by Li depletion in non-magnetic models (Somers & Pinsonneault

2014), or by the inclusion of rotational mixing (Somers & Pinsonneault 2015a), but naturally emerges from a range of spot properties (i.e. a range of radii) at fixed mass and age during the pre-MS. Furthermore, my predictions are consistent both with the radius data for active eclipsing binaries, and with the anomalous colors of young active stars in the Pleiades, suggesting that my models can reproduce observational results with realistic spot properties, and that magnetic activity continues to inflate rapidly rotating stars on the ZAMS and beyond. Collectively, these results indicate that the differential magnetic properties of stars (correlated with rotation) lead to a dispersion in radius at fixed mass and age, which emerges on the early pre-MS and persists on the main sequence while stars remain rapidly rotating.

This conclusion is broadly consistent with claims of radius dispersion in pre-MS (Cottaar et al. 2014), and young main sequence (Jackson, Jeﬀries & Maxted

2009; Jackson & Jeffries 2014a) open clusters, but would appear to be in conflict with the uniform saturation of magnetic proxies among young stars, which could be interpreted to mean that spot properties are identical at fixed mass on the pre-MS. However, I note that saturation is merely an empirical statement about chromospheric and coronal heating, and that a direct correlation with spot properties

185 has not yet been proven (see Section 5.6.3). Alternatively, saturation may indicate that other phenomenon, such as mass accretion (e.g. Littlefair et al. 2011), play a role in generating the pre-MS radius dispersion needed to explain the Li data (see the discussion in Somers & Pinsonneault 2015a), though this scenario still relies on activity to explain the radii of eclipsing binaries and the colors of rapidly rotating

Pleiads.

A radius dispersion in pre-MS clusters have several signiﬁcant implications.

First, I have shown that spots make stars appear cooler, and as a result heavily spotted stars coincide on the HR-diagram with younger, less massive stars which are weakly spotted or pristine (Section 5.4.1). The range in masses at ﬁxed HR diagram location can reach a factor of two, while the range in ages can be signiﬁcantly higher

(up to ∼ 3× above 0.5M , and larger for lower masses). Consequently, pre-MS ⊙ cluster ages inferred by matching isochrones to the locus of HR diagram data will be erroneously low. The magnitude of the age error may match the above-quoted correction factors if stars are uniformly spotted, or may be somewhat less if a range of spot filling factors blurs out an otherwise sharper stellar locus. Furthermore, if stars of equal age have differential spot properties, the inferred ages can differ substantially, leading to the appearance of an intra-cluster age spread. Appreciating the magnitude of this effect is important for inferring the true age spreads within young clusters.

186 As noted in Section 5.5, the lithium depletion boundary technique also systematically under-estimates the ages of clusters whose members are heavily spotted, by up to 15% in the most extreme case I considered. As many previous young cluster studies do not consider these magnetic effects, my findings necessitate an upward revision of the age scale for clusters between 1 and 100 Myr. The magnitude of this revision is difficult to anticipate, as the required corrections for each individual study depend strongly on the true age of the cluster and on the mass range probed, as well as the distribution of spot properties among stars in each cluster. A full re-analysis of young cluster CMDs in the context of magnetic models is beyond my current purpose, but would undoubtedly have important implications for, among other things, the duration of circum-stellar disc lifetimes and the accretion timescales of proto-stars, both of which are inferred through cluster ages.

Second, Fig. 5.3 shows that among spotted stars, the least massive require the largest correction factors. Consequently, in a mono-aged cluster the lowest mass objects will appear to have the youngest isochronal ages, and the hottest stars will appear to have the oldest isochronal ages. Thus in the presence of spots, clusters should show a trend in inferred age with temperature. This precise behavior was noted in the young β Pic association by Malo et al. (2014): stars with Teﬀ > 3500 K appeared to be 15 Myr on average, while cooler stars appear to be 4.5 Myr on average, both of which are younger than the generally accepted age of ∼ 21 ± 4 Myr

187 (Soderblom et al. 2014). They showed that this discrepancy could be ameliorated through the inclusion of strong magnetic ﬁelds in the cooler stars, and weaker

ﬁelds in the hotter stars. Alternatively, consistent spot properties at all masses will naturally impact the age determinations of lower mass stars to a greater degree, and thus can produce just such a disagreement. The relative isochronal ages of cool and warm stars depends on the relative spot properties of the two populations, and therefore provides an empirical constraint on the surface magnetic ﬁelds of pre-MS stars.

Finally, the systematic under-estimation of the masses of stars will bias measurements of the initial mass function (IMF) which employ pre-MS clusters. In particular, the average low mass star will require a signiﬁcant upward revision of its mass, ultimately implying a more top heavy IMF, and a lower number of brown dwarfs. The magnitude of this eﬀect has been discussed in Stassun et al. (2014), and

I refer the reader there for a more detailed account.

5.6.2. Literature comparison

Heavy spot coverage can enhance stellar radii by up to ∼10% during the pre-MS at

fixed age, due in part to the slower contraction and in part to the impact of spots on the pressure conditions at the photosphere. Once on the MS, the peak radius anomalies are mass dependent, with partially-convective stars generally more inflated than fully-convective stars. This pattern is unsurprisingly similar to the findings

188 from Spruit & Weiss (1986), with the exception that my models predict a decrease in ∆R at fixed spot properties between 0.4-0.8M , while Spruit & Weiss predict flat ⊙ ∆R in this range. The origin of this discrepancy is unclear, but is probably related to different input physics which cause a substantial shallowing of the mass-radius relationship around 0.7M . ⊙

Assuming equipartition between the magnetic and gas pressures in the spotted regions (i.e. Pspot,mag = Pspot,gas) and pressure equilibrium between the spots and the photosphere, my models imply magnetic ﬁeld strengths of 2.2, 1.6, and 1.1 kG at the ZAMS for 0.2, 0.5, and 1.0M , respectively. This is generally consistent with ⊙ the 1-3 kG range reported for empirical measurements (e.g. Cranmer & Saar 2011).

Combined with my spot coverage factor of 0.5, the surface averaged fields = ∗ ∗ 0.6-1.1 kG. This value is similar to the findings of Mullan & MacDonald (2001), who require surface strengths of less than 1 kG to inflate fully convective stars by ∼3-4%.

They are however somewhat lower than the 2.0-6.0 kG surface ﬁelds required by

Feiden & Chaboyer (2013, 2014) to match the observed radii, Teff s, and luminosities of several eclipsing binaries. This quantitative discrepancy is not surprising, as their studies make significantly different assumptions about magnetic field morphology. I note however that the mass dependence they find is qualitatively similar to ours; stronger magnetic fields are required to inflate fully convective stars relative to solar type stars, consistent both with the pattern revealed in Fig. 5.1, and the ∗ ∗

189 estimates given above. This suggests that greater inﬂation at higher mass is a generic prediction of magnetic stellar models.

In Section 5.5, I found that spot-induced errors on cluster ages derived from the

LDB technique are large compared to systematic uncertainties in the model inputs

(∼15%, compared to <8%; Burke et al. 2004). However, the errors I derived are somewhat lower than the ∼20% age errors reported by Jackson & Jeﬀries (2014b), who considered the impact of similar spot coverage on lithium depletion in their polytropic models. These results are in reasonably good agreement, considering the numerous physical inputs required in stellar modeling, but it is interesting to consider the source of this discrepancy. In a polytrope model, the internal gradient

∇ is assumed to be ∇ad throughout the entirety of convective regions. While this is a good assumption for the vast majority of the interior of fully-convective stars, it is less exact in the thin layer at the surface where the gradient becomes super-adiabatic (e.g. Kippenhahn & Weigert 1990). As it happens, starspots chiefly reside near this super-adiabatic layer, and so their effect on the structure of the star will be dependent on the adopted physics of this region. As discussed in Section 5.3, changing the temperature at the boundary of a convective zone leads to a linear shift in P throughout the region. Starspots are placed at this outer boundary in the polytropes of Jackson & Jeffries (2014b), so the resulting L shift in the center is approximately equal to the L blocked at the surface. In my standard stellar models, the temperature shift first propagates through the super-adiabatic layer before

190 impacting the convective envelope, leading to a somewhat higher central value of

L. This in turn impacts the Li depletion age as a function of mass, and thus the inferred ages of clusters. However, I note that the diﬀerent quantitative results may derive simply from diﬀerent choices of input physics (Burke et al. 2004).

5.6.3. Observational and physical concerns

My approach in this paper has been to adopt a ﬁxed set of spot parameters (i.e. fspot = 0.5, xspot = 0.8 regardless of age), and derive their eﬀect on stellar properties.

While such limits are useful, the ideal analysis would tie the temperature, covering fraction, and distribution of spots to the (evolving) structure and properties of their host stars. This physical approach would permit a realistic population model of spotted stars in young clusters, which is desirable for understanding the eﬀect of spots on observables during the pre-MS and MS. Given the present state of the

ﬁeld, such a model awaits superior understanding of the underlying mechanisms governing the behavior of stellar magnetism. I highlight in this section some key outstanding questions whose resolution will improve the capacity of stellar theory to model magnetic ﬁelds in young clusters.

Spot depth

Spots extend to a ﬁnite depth within the envelopes of stars, but exactly how deep they penetrate, and the details of their subsurface morphology, remain a topic of debate (e.g. Rempel & Schlichenmaier 2011). In my models, I have assumed that

191 Tamb − Tspot is constant as a function of depth, a criterion which leads to shallow spots3. To assess how this assumption impacts my results, I compare my base case to models which assume a constant temperature contrast with depth (Tamb/Tspot = constant). This criterion implies very deep spots, since the temperature contrast at the surface is maintained at the base of the surface convection zone.

The comparison appears in Fig. 5.8. The top panel shows the radius anomalies resulting from the two depth treatments at the ZAMS, and the bottom panel reports the difference between the curves. Deep spots lead to greater radius inflation with a maximum difference of ∼1.4%. This is a small, but non-negligible effect relative to the overall radius anomaly (∼10%). This suggests that the global impact of spots is chiefly due to their effect on the very outer regions of the star, but is also moderately sensitive to flux redistribution in the envelope. Constant spot contrast is an extreme limiting case, as it would imply an extremely strong magnetic field (tens to hundreds of MG), and a strong magnetic field gradient, near the base of the convection zone.

I therefore expect the true diﬀerence to be signiﬁcantly less than seen in Fig. 5.8.

Despite the marginal diﬀerences, shallow and deep spots produce quantitatively similar results, and I therefore conclude that depth eﬀects do not impact my results.

3 As an example, a surface spot contrast of xspot = 0.8 in a 1M⊙ model reduces to less than a 2% eﬀect at about 3 pressure scale heights below the photosphere, being in the outer 10−9 and 10−3 in mass and radius, respectively.

192 Magnetic saturation

A key outstanding question in the study of stellar magnetism is the physics underlying magnetic saturation. Numerous studies have determined that once stars are rotating faster than a critical Rossby number (∼0.1; e.g. Noyes, Weiss & Vaughan

1984; Pizzolato et al. 2003), classical magnetic ﬁeld proxies cease to increase with faster rotation. This observation suggests a saturation in the strength of magnetic torques and chromospheric/coronal heating, which are the physical processes that generate the observed proxies. The cause of this saturation remains unclear, but is probably related to a change in the properties of the stellar dynamo when the rotation period becomes substantially shorter than the convective overturn timescale

(i.e. low Rossby number).

Observations of young clusters indicate that the vast majority of pre-MS stars are in the saturated regime (Preibisch et al. 2005; Argiroffi et al. 2013). If all magnetic effects, such as spots, saturate along with the classical proxies, then one would expect uniform spot-properties at fixed mass on the pre-MS, and thus no dispersion in stellar properties due to magnetic activity. However, there is at present no convincing demonstration that spots saturate along with magnetic proxies, and some studies have in fact found evidence that the amplitude of spot modulation varies substantially in the saturated regime (e.g. O’dell et al. 1995; McQuillan et al.

2014), perhaps suggesting disparate spot properties. Furthermore, a uniform spot coverage does not accord with the evidence from Li observations, which indicate

193 a rotation-correlated dispersion in radii at ﬁxed mass and age during the pre-MS

(Section 5.5). Uncovering the spot properties of pre-MS stars has been hampered by observational issues, primarily the substantial degeneracies between ﬁlling factor, the surface distribution of spots, and their temperature relative to the photosphere

(Jackson & Jeﬀries 2013). Nevertheless, the connection (or lack thereof) between saturated proxies and spot properties is an essential component of a complete theory of stellar magnetic activity, and must be understood in order to appreciate the impact of spots on the pre-MS.

Plages

Another confounding eﬀect is the presence of plage regions in stellar photospheres.

Plages are the hot analogs of starspots, and radiate light at higher frequencies than the un-spotted surface. This enhanced emission may counteract the total flux suppression caused by spots, and thus reduce the magnitude of the perturbations to stellar parameters that I have outlined in Section 5.3-5.5. The size of this correction relies on the difference between the plage enhancement and spot decrement to the total irradiance, which is likely dependent on mass and rotation among other parameters. Furthermore, plage regions provide a source of anomalous blue emission, which has significant effects for the SEDs of active stars (e.g. Stauffer et al. 2003).

These regions are poorly understood at present, so a deeper understanding of their origin, and their correlations with spots on pre-MS and ZAMS stars, is necessary.

194 Spot evolution

Finally, in my models I have assumed that spots appear before the deuterium birth line, and persist at the same magnitude for the entirety of the approach to the MS.

In reality, spot properties likely change over contraction timescales as the rotation rate and convective properties of stars evolve, as well as on shorter timescales due to the appearance and dissolution of individual spots. A fixed filling factor can approximate the time-averaged effect of spots, but may miss crucial physics occurring on sub-thermal timescales. Furthermore, I have assumed a single temperature contrast between the spotted and ambient regions. In fact, spot temperatures appear to correlate with mass (e.g. Berdyugina 2005), and may evolve during the pre-MS due to the changing pressure, density, and rotation rate of contracting stars. A superior treatment of these dynamic quantities is necessary for robust spot predictions.

The heliographical distribution of spots may also play a role in determining the efficiency of energy redistribution. For instance, if a large number of small spots are evenly distributed throughout the photosphere, then the entire un-spotted surface will be in close proximity to magnetic regions. By contrast, if the star is populated by a few large spots, then certain ambient regions will be much farther from the closest magnetic region. The average distance of the photosphere from spots could impact the efficiency and uniformity of flux redistribution, and might therefore change the structural effect of starspots. More sophisticated (2-D or 3-D)

195 spotted models may help shed light on the role of spot distribution on surface and sub-surface structure.

5.7. Summary

Through the inclusion of spots in my stellar models, I have explored the impact of a non-uniform surface on the radii, Teﬀ , luminosity, inferred masses and ages, colors, and Li abundances of pre-MS and ZAMS stars.

I ﬁnd that a 50% coverage of 80% temperature contrast spots (an eﬀective blocking area of 30% black spots) increases the radii of all stars on the pre-MS by

8-12%, and by a mass-dependent amount on the MS, which increases from ∼3% in fully-convective stars to ∼10% at 1.2M . This behavior is a product of both the ⊙ impact of spots on the central luminosity of stars, and on the ﬁne structure of the mass-radius relationship. Moreover, these anomalies are similar in magnitude to claimed radius anomalies in detached eclipsing binary systems, suggesting that spots may play a role in the well known radius inﬂation of young, active low mass stars.

In my models, these radius anomalies are accompanied by corresponding decrements to the luminosity, which are large in fully-convective stars and small in stars with thin envelopes, and to the Teff , which are weakly sensitive to mass. These perturbations are physical, and displace stars from their fiducial location in the HR diagram. This effect has several implications:

196 1) Masses derived from spotted stars using pre-MS isochrones will be systematically low by up to 2× at 3 Myr, and ages will be systematically young by up to 10× for the lowest mass stars. I quantify the resulting errors as a function of mass and age, and present correction factors for spotted stars.

2) Spotted stars will be preferentially displaced towards the upper right of the standard HR diagram; in particular, they may naturally appear above the un-spotted deuterium birth line. Detecting spot modulation in this region of the HR diagram therefore presents a direct test of the prevalence of spots on pre-MS stars.

3) A range of masses and ages may be present at ﬁxed HR-diagram location, and low mass stars will generally appear younger at ﬁxed age compared to higher mass stars. Interpreters should be cautious about relaying the fundamental parameters of stars without considering the impact of activity.

4) The observed colors of stars will reﬂect an area-weighted sum of the emission from the hot and cold regions. I ﬁnd that spotted stars appear bluer than un-spotted stars in short-wavelength bandpasses, and redder in long-wavelength bandpasses, consistent with observations of (spotted) stars in the Pleiades and other young clusters. Furthermore, the extent of the blue-ward shift depends strongly on the average temperature of the spots, which may help to break observational degeneracies between total spot coverage, surface distribution, and temperature.

197 Finally, I examined the impact of spots on pre-MS lithium depletion in two regimes: fully-convective stars, and in G and K dwarfs. For the former category, the main interest is measuring the ages of young clusters through the lithium depletion boundary technique. I found that an eﬀective blocking area of 30% can increase the inferred age by ∼ 10-15% relative to the true age. This bias exacerbates the tension between LDB ages and main sequence turn-oﬀ ages in young clusters such as the Pleiades. The errors I derive are less severe than the results of Jackson &

Jeﬀries (2014b), with the diﬀerence potentially stemming from the treatment of the super-adiabatic region in the two modeling approaches. For G and K dwarfs,

I find that spots inhibit the destruction of Li during the pre-MS by altering the temperature at the base of the convection zone. A range of spot filling factors in my pre-MS models leads to varying degrees of Li destruction and thus a dispersion at fixed mass and temperature on the ZAMS, consistent with observational studies of young clusters. This implies that clusters during the epoch of Li destruction

(∼ 5 − 15 Myr) are differentially affected by magnetism, imprinting a radius dispersion of ∼ 5 − 10% at fixed mass.

198 3.0 Unspotted A) 2.0 50% Spotted

M = 1.20M ⊙

) 1.0 ⊙ M = 1.00M ⊙ R (

M = 0.70M ⊙ s 0.5 u

i M = 0.50M ⊙ d a

R M = 0.30M ⊙

M = 0.20M ⊙

M = 0.12M ⊙ 0.1

) 20 % ( 1.2

s 10

u 1.0 i

d 0.5 5 0.7 a R 0.3 0.2 0.12 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 log(Age/Myr)

12 f =0 B) spot Zero Age Main Sequence 10 fspot =1/6 ) fspot =1/3 % 8 ( fspot =1/2 s 6 u i d

a 4 R

2 0

10 R 0

Teff −10

Radius −20 L Luminosity fspot = 1/2 Teff −30

Fractional Difference (%) 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Mass (M ⊙)

f =0 C) 1.2 spot fspot =1/2 1.0 )

⊙ 0.8 R (

u 0.6 i d a

R 0.4

0.2

0.0 0.04 0.00 −0.04 −0.08 −0.12 Radius - Mass 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Mass (M ⊙)

Fig. 5.1.— Impact of starspots on radii, luminosities, and Teff. The impact of spots on radii, luminosities, and Teff s of stars in the mass range 0.1-1.2M . A) The ⊙ time evolution of the radius of un-spotted (solid) and spotted (dashed) models of various masses are shown in the top. The bottom shows evolution of the difference in radius between the spotted and un-spotted models for fully convective (dotted) and partially convective (solid) models. ∆R steadily grows during the early pre- MS, before stabilizing at the entrance to the MS. B) The top shows zero-age main sequence radius anomalies across my mass range, for three different values of fspot. The bottom plots the luminosity and Teff anomalies resulting from the case of fspot = 0.5, respectively shown in blue and red. C) The mass-radius relationship for un- spotted (light) and spotted (dark) models. The bottom panel shows that the mass- dependence of the structure of the relationship is affected by spots; this leads to the strong mass dependence of ∆R shown in panel B.

199 Fig. 5.2.— Impact of starspots on the mass-radius relationship. Left: The mass- radius relationship for un-spotted (black) and spotted (blue) models at the ZAMS, and the 10 Gyr relation for un-spotted models (red dashed). These are compared to data from eclipsing binaries (see the text for references). Right: The same as the left, but detrended against the ZAMS, un-spotted isochrone. Dotted purple represents the sum of age and spot eﬀects. Two-thirds of the data lie within the spotted models, and all but one lie within the combined eﬀects of activity and age.

200 20 Age Correction Factors Mass Correction Factors 1 Myr 3 Myr 10 1.20 10 Myr 1 1.20

0.90 0.90 ) ⊙ 0.70 Age Correction Factors

L 0.70

( 5 y

t 0.50 i 0.50 s o n i 0.35 3 0.1 0.35 m u L t =1 Myr 0.20 2 0.20 t =3 Myr

Mass/Age Correction Factors t =10 Myr Un-spotted tracks t =1 Myr Un-spotted isochrone (3 Myr) t =3 Myr 50% spotted tracks 1 t =10 Myr 0.01 Mass Correction Factors 50% spotted isochrone (3 Myr) 5500 5000 4500 4000 3500 3000 2500 1.4 1.2 1.0 0.8 0.6 0.4 0.2

Effective Temperature (K) Inferred Mass (M ⊙)

Fig. 5.3.— Impact of starspots on the masses and ages of young stars. Left: Un- spotted (red) and spotted (blue) evolutionary tracks and 3 Myr isochrones for a range of masses. Spots displace stars to the lower right (cooler and less luminous) at fixed age. As a result, a dispersion in luminosity at fixed Teff develops across the mass function, and spotted stars appear less massive and younger when interpreted with un-spotted isochrones. Right: Correction factors which describe the errors incurred when measuring the mass and age of spotted stars with un-spotted isochrones. At a given mass (x-axis) and age (red/orange/yellow) inferred from un-spotted isochrones, the y-axis reports the necessary corrections. For example, a heavily spotted star which appears to have a mass of 0.5M and an age of 3 Myr will have respective ⊙ corrections (green crosses) of 1.5 (implying a true mass of 1.5 × 0.5 M ∼ 0.75 M ), ⊙ ⊙ and 2.7 (implying a true age of 2.7 × 3 Myr ∼ 8 Myr).

201 Fig. 5.4.— Impact of starspots on young cluster color-magnitude diagrams. Using empirical bolometric corrections, my un-spotted (red) models are directly transformed into three diﬀerent color coordinates using a two-zone spot model. In color coordinates, the spotted models do not appear as cool due to the contribution to short wavelength emission from the un-spotted photosphere. The spot models appear bluer when shorter wave-lengths are used, due to the progressively stronger sampling of the Wein tail of the blackbody function. The bottom right reproduces the models in the theoretical HR diagram for comparison.

202 Fig. 5.5.— The effet of spot temperature on color-magnitude diagrams. The effect of spot filling factor and temperature on the color-magnitude locations of spotted models with the same spot blocking parameter αspot. Darker spots cause models to appear bluer, as the hot, un-spotted photosphere progressively dominates the total emission. The bottom right panel shows surfaces of constant flux redistribution in the plane of fspot vs. xspot, and the colored dots represent the different models shown in the other three panels.

203 0.075 25 f =0 2.2 spot 0.080 f =1/6 Age of 95% Fully Convective spot lithium depleton 0.075 0.090 Boundary f =1/3 2.0 spot 0.080 20 0.100 fspot =1/2

0.090 0.120 1.8 0.100 0.140 15 0.120 1.6 0.180 0.140 0.220 0.260 1.4 0.180 10 log(Age/Myr) 0.300 0.220 0.350 0.260

0.400 Fractional Age Error (%) 1.2 0.300 0.450 0.500 5 0.350 0.400 1.0 fspot =0 0.450 f =1/2 0.500 spot 0 0.8 −3.0 −2.5 −2.0 −1.5 −1.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Luminosity (L ⊙) log(Age/Myr)

Fig. 5.6.— Impact of starspots on lithium depletion cluster ages. The impact of spots on the accuracy of cluster ages obtained through the lithium depletion boundary technique. Left: Lines of constant spot filling factor denote the age at which 95% of the primordial Li is destroyed from models with a given luminosity. Spotted models lie above the un-spotted case, demonstrating that spots delay the onset of Li destruction, thus biasing determinations towards younger ages. Right: The fractional error on the age determination as a function of true age, for three different filling factors. The errors are approximately half of the spot blocking parameter αspot (0.3, 0.2, and 0.1 respectively for purple, blue, and red) up to ∼60 Myr, and drop off thereafter.

204 3.5 Initial A(Li)

3.0 fspot =0.5

2.5

2.0 fspot =0.0 A(Li)

1.5

Age = 125 Myr 1.0 Prot < 1 day

1 day < Prot < 3 days

0.5 3 days < Prot < 6 days

6 days < Prot 6500 6000 5500 5000 4500 4000 Effective Temperature (K)

Fig. 5.7.— How starspots inhibit the depletion of lithium. The impact of pre-MS spots on Li destruction in partially convective stars. Higher coverage results in a lower pre-MS depletion rate, and thus a higher abundance at the ZAMS. The resulting range and temperature dependence is similar to the Li pattern observed in the Pleiades (grey circles).

205 12 Tamb−Tspot = const.

10 Tamb/Tspot = const.

8 ) % (

s 6 u i d a

R 4

0 1.5 1.2 0.9 0.6 0.3 Difference (%) 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Mass (M ⊙)

Fig. 5.8.— The impact of spot depth on radius inflation. The impact of spot depth on radius inflation. The black and red dashed lines represent shallow and deep spot treatments, respectively. The top shows absolute anomalies, and the bottom shows the difference between the anomalies as a function of mass. The difference reaches at most ∼1.5%, and thus impacts detailed calculations, but not the conclusions of this paper.

206 Inferred Mass Mass Correction Age Correction (M ) Factors Factors ⊙

fspot = 1/6 1Myr 3Myr 10Myr 1Myr 3Myr 10Myr

0.20 1.263 1.275 1.296 3.589 2.311 1.881 0.22 1.249 1.263 1.289 3.280 2.191 1.835 0.24 1.234 1.251 1.281 3.010 2.084 1.790 0.26 1.220 1.239 1.270 2.777 1.986 1.740 0.28 1.205 1.226 1.263 2.561 1.893 1.704 0.30 1.194 1.214 1.255 2.397 1.818 1.668 0.35 1.175 1.194 1.236 2.120 1.686 1.597 0.40 1.163 1.181 1.217 1.939 1.600 1.535 0.45 1.151 1.172 1.198 1.782 1.532 1.495 0.50 1.137 1.160 1.178 1.648 1.464 1.449 0.55 1.123 1.148 1.155 1.535 1.407 1.406 0.60 1.106 1.135 1.133 1.434 1.357 1.368 0.65 1.095 1.119 1.111 1.383 1.312 1.332 0.70 1.093 1.109 1.090 1.360 1.286 1.295 0.80 1.093 1.097 1.059 1.335 1.264 1.254 0.90 1.092 1.085 1.036 1.313 1.242 1.196 1.00 1.089 1.074 1.013 1.294 1.226 1.163 1.10 1.088 1.063 1.002 1.286 1.208 1.135 1.20 ... 1.052 0.994 ... 1.193 1.127

(cont’d) Table 5.1. Mass and Age Correction Factors

207 Table 5.1—Continued

Inferred Mass Mass Correction Age Correction (M ) Factors Factors ⊙

fspot = 1/3 1Myr 3Myr 10Myr 1Myr 3Myr 10Myr

0.20 1.593 1.633 1.710 7.349 4.321 3.419 0.22 1.548 1.595 1.681 6.412 3.937 3.244 0.24 1.510 1.560 1.650 5.686 3.622 3.083 0.26 1.477 1.529 1.620 5.104 3.358 2.938 0.28 1.450 1.501 1.592 4.641 3.139 2.813 0.30 1.427 1.477 1.564 4.268 2.958 2.702 0.35 1.388 1.435 1.499 3.606 2.635 2.487 0.40 1.355 1.401 1.434 3.114 2.399 2.291 0.45 1.321 1.370 1.375 2.714 2.204 2.132 0.50 1.284 1.338 1.321 2.383 2.034 1.999 0.55 1.248 1.303 1.271 2.125 1.887 1.883 0.60 1.220 1.271 1.224 1.949 1.772 1.781 0.65 1.206 1.243 1.184 1.856 1.689 1.694 0.70 1.202 1.223 1.148 1.801 1.639 1.627 0.80 1.195 1.191 1.092 1.723 1.571 1.524 0.90 1.183 1.161 1.046 1.660 1.513 1.403 1.00 1.182 1.132 1.017 1.644 1.460 1.327 1.10 1.182 1.110 1.000 1.640 1.428 1.280 1.20 ...... 0.984 ...... 1.279

(cont’d)

208 Table 5.1—Continued

Inferred Mass Mass Correction Age Correction (M ) Factors Factors ⊙

fspot = 1/2 1Myr 3Myr 10Myr 1Myr 3Myr 10Myr

0.20 2.005 2.093 2.156 12.819 7.453 5.968 0.22 1.931 2.023 2.074 11.039 6.649 5.490 0.24 1.866 1.961 2.000 9.639 5.993 5.076 0.26 1.812 1.904 1.931 8.526 5.453 4.721 0.28 1.765 1.854 1.867 7.630 5.003 4.413 0.30 1.725 1.809 1.808 6.888 4.627 4.139 0.35 1.643 1.719 1.680 5.523 3.928 3.591 0.40 1.573 1.644 1.571 4.545 3.419 3.177 0.45 1.502 1.575 1.478 3.789 3.017 2.864 0.50 1.439 1.510 1.401 3.229 2.692 2.609 0.55 1.390 1.454 1.333 2.841 2.450 2.407 0.60 1.352 1.405 1.275 2.571 2.270 2.245 0.65 1.329 1.362 1.224 2.409 2.134 2.110 0.70 1.313 1.325 1.179 2.291 2.038 1.992 0.80 1.286 1.265 1.105 2.134 1.893 1.790 0.90 1.285 1.211 1.051 2.108 1.772 1.612 1.00 1.286 1.177 1.018 2.108 1.725 1.491 1.10 ... 1.150 0.994 ... 1.703 1.454 1.20 ...... 0.968 ...... 1.480

Note. — Mass and age correction factors for three spot ﬁlling factors at various inferred masses and ages. Correct useage is detailed in Section 5.4.1. Ellipses indicate inferred mass and age points outside of the interpolation grid.

209 Chapter 6: Lithium Depletion is a Strong Test of Core-Envelope Recoupling

6.1. Introduction

In Chapter 2, I detailed the observational pattern of open cluster Li abundance as a function of age, and noted that Li appears to be gradually depleted from solar-type stellar photospheres during the main sequence (e.g. Soderblom et al.

1993a; Thorburn et al. 1993; Jones et al. 1999; Lodders 2003). These observational features can be interpreted as a consequence of rotationally-induced mixing pumping

Li-depleted material from the deep interior into the convective envelope. This explanation is attractive, as the energetics of stellar rotation are suﬃcient to produce the required mixing, and the observed dispersion of natal rotation rates naturally predicts varying rates of mixing, and thus dispersions in Li at ﬁxed age. Moreover, the rate of rotationally induced Li depletion is generically predicted to decline with age, consistent with the data. However, previous rotational mixing calculations have generally relied on incomplete treatments of internal angular momentum transport, which cannot reproduce the observed rotational evolution of open clusters. In this chapter, I propose to incorporate into my mixing models a more sophisticated

210 treatment of angular momentum transport, and determine the consequences for Li depletion.

Solar-type stars are born rapidly rotating (Prot = 0.1 − 10 d), and subsequently spin down by factors of 5-500 before exhausting their hydrogen cores. Substantial work has gone into understanding this spin down, and over the past decades a consistent picture has emerged. Mass driven from the surface by magnetized winds robs the outer convective zone of angular momentum (Weber & Davis 1967), causing the envelope to decouple from the radiative interior and spin down. This process produces strong diﬀerential rotation between the slowly-rotating envelope and the rapidly-rotating core. Once on the main sequence, transport mechanisms drive angular momentum out of the core into the envelope, leading the radiative and convective regions to converge in rotation over some timescale set both by underlying physics of the transport processes and the unique parameters of the individual star

(Pinsonneault et al. 1989). Thus, for both single stars and members of multiple systems which are adequately separated from their companions, two main eﬀects govern angular momentum evolution, 1) loss at the surface, 2) transport between interior layers.

The nature of these internal transport mechanisms is an important open question in stellar physics. The only securely-known processes are those resulting from hydrodynamic instabilities (for an extensive review, see Talon 2008), such as meridional circulation (Eddington 1929; Sweet 1950), associated baroclinic

211 instabilities (Zahn 1992), shear instabilities (Zahn 1974; Maeder & Zahn 1998), and the Goldreich-Schubert-Fricke mechanism (Goldreich & Schubert 1967; Fricke 1968).

However, early rotation models incorporating solely hydrodynamic eﬀects predict strong diﬀerential rotation at the solar age (e.g. Pinsonneault et al. 1989; Chaboyer,

Demarque & Pinsonneault 1995), and can not explain the near-solid-body solar rotation proﬁle inferred from helioseismology (Libbrecht 1988; Schou et al. 1998).

Furthermore, they could not simultaneously explain the evolution of the rapid and slow rotating branches of open clusters from the pre-main sequence to the solar age

(Krishnamurthi et al. 1997).

By contrast, recent rotation models have found success in predicting the evolution of open cluster rotation rates by supposing that angular momentum transport is in fact stronger than the predictions of pure hydrodynamics (e.g.

MacGregor & Brenner 1991; Denissenkov et al. 2010; Gallet & Bouvier 2013;

Lanzafame & Spada 2015). These papers have followed two general patterns.

Some use the so-called “two-zone model”, which treats the convective envelope and radiative core as solid body regions, whose rotation rate can differ from one another, but who converge in rotation with a specified core-envelope coupling timescale τCE (e.g. MacGregor & Brenner 1991; Gallet & Bouvier 2013). Others assume the existence of a background source of angular momentum transport that operates in conjunction with hydrodynamics (Denissenkov et al. 2010). As both internal magnetic fields (e.g. Gough & McIntyre 1998; Eggenberger et al.

212 2005) and gravity waves (Press 1981; Ando 1986; Charbonnel & Talon 2005) are known to operate in stellar interiors, they are considered strong candidates for the background mechanism. Both techniques conclusively demonstrate that one or more additional sources of internal angular momentum transport are required to explain the rotational evolution of stars.

For all their success, these recent papers have predominately focused on rotation, with less attention given to the impact of these additional transport processes on an important consequence of angular momentum evolution: rotational mixing (e.g. Pinsonneault 1997). Rotational mixing occurs in concert with hydrodynamic angular momentum transport, since the hydrodynamic instabilities discussed above physically transport material from one layer to another, carrying the speciﬁc abundances of chemical species within the transported parcel of mass

(Pinsonneault et al. 1989; Chaboyer, Demarque & Pinsonneault 1995; Li et al. 2014).

Rotational mixing can alter the observed chemical abundances of stars by mixing nuclear-processed material into the surface layers, and can extend the lifetimes of stars beyond the predictions of standard stellar theory by replenishing the nuclear core with fresh hydrogen. Core-envelope recoupling would naturally impact these processes, as it fundamentally alters the angular momentum history of stars. It is worth noting though that the additional transport mechanisms typically invoked

(magnetic ﬁelds, gravity waves) do not directly mix chemical material, but instead

213 modulate the internal angular momentum proﬁle, which in turn inﬂuences the rate of hydrodynamic mixing.

Given the important consequences of mixing, and the patent need for non-hydrodynamic angular momentum transport in stellar interiors, I investigate the impact of core-envelope recoupling on mixing. I do this by including in my fully-consistent evolution code a constant diffusion coefficient (D0), which increases the diffusion of angular momentum at all layers in the radiative region by a fixed amount (Denissenkov et al. 2010), but does not mix chemical material (see §6.2.3 for details). This approach is better suited to study mixing than the two-zone model, as Li depletion depends sensitively on the degree of differential rotation in the outer layers of the core, structure which two-zone calculations do not resolve.

I also introduce a new method for calibrating main sequence mixing models, which seeks to mollify certain inconsistencies between typical mixing calculations and their empirical constraints. The most egregious example is the spuriously strong Li depletion predicted by mixing models during the pre-main sequence: evolutionary models predict that solar-mass rapid rotators signiﬁcantly deplete Li during their ﬁrst 20 Myr, whereas the solar-mass stars in the Pleiades are no more

Li-poor than their slowly rotating counterparts (e.g. Ventura et al. 1998; Somers

& Pinsonneault 2015b). I address this issue by initializing the Li abundances on the zero-age main sequence. A second example are complications arising from interactions between young stars and their circum-stellar accretion disks, which

214 have profound consequences for the angular momentum evolution of T Tauri stars.

As such stars are the anchor for the initial rotation distribution of population models, the obfuscating eﬀects of disk-locking must be dealt with. Finally, previous mixing exercises have typically used the Sun as the calibrator both for the rate of angular momentum loss, and for the strength of chemical mixing. However, pegging a physical model to a single point could be misleading, if that datum were an outlier. Calibrating instead to an ensemble average will be more robust, because a distribution of initial conditions produces a distribution of Li depletion factors. In order to produce informative models of main sequence Li depletion, I address each of these complications in my calibration procedure (§6.4).

I incorporate these techniques into my rotating evolution code, and explore the mixing signature arising from core-envelope recoupling driven by enhanced angular momentum transport. As I will show, these “hybrid” models, as I call them, can simultaneously replicate both the open cluster rotation and Li depletion pattern far better than other models of stellar evolution. Furthermore, I reveal a strong mass dependence in the rate of core-envelope coupling, which has been seen previous in rotation, and which I demonstrate for the ﬁrst time using lithium.

This chapter is organized as follows. I describe my evolutionary models, and the adopted calibration data sets, in §6.2. I then introduce both the rotation and mixing behavior of the hybrid models, and compare/contrast them with traditional limiting cases of angular momentum transport (§6.3). In §6.4, I lay out my calibration

215 scheme. I then calibrate the efficiency of core-envelope recoupling in solar-mass stars using open cluster rotation data (§6.5.1) and the efficiency of mixing in several mass ranges using open cluster Li data (§6.5.2). From these results, I derive the implied core-envelope coupling timescales (§6.5.3), and discuss implications for gravity waves and magnetic fields as transport agents. In §6.6, I compare my models to the internal rotation profile of the Sun (§6.6.1), and discuss the possible existence of cosmic variance in open cluster properties (§6.6.2). Finally, §6.7 concludes.

6.2. Methods

In §6.5, I constrain the rate of angular momentum loss, the required strength of the constant diffusion coefficient, and the efficiency of mixing in a range of stellar models. To do this, I employ spot-modulation rotation periods and Li abundances from a series of open clusters, and utilize my stellar evolution code.

6.2.1. Rotation data

For this chapter, I will compare my rotational models to statistical bounds of the open cluster rotation distribution as a function of age. While the entire distribution could in principle be modeled, this technique is far more sensitive to sample contamination, such as cluster non-member backgrounds, and binary components whose angular momentum evolution has been inﬂuenced by tidal forces from their companions. By taking average moments of a full distribution, much of this

216 contamination can be “averaged out”, and a realistic picture of the evolution of the distribution emerges.

The rotation comparison data comprises the evolution of the 25% (slow),

50% (median), and 90% (fast) bounds of the 0.9 − 1.1 M open cluster rotation ⊙ distribution, as reported by Gallet & Bouvier (2015). For all but two of the clusters,

I adopt the ages listed in table 1 of Gallet & Bouvier (2015), and assume a 10% age error, following the recommendation of Soderblom (2010). The exceptions are the

Hyades and Praesepe for which a recent study has reported an age of ∼ 800 Myr

(Brandt & Huang 2015), distinct from the ∼ 600 Myr values adopted by Gallet

& Bouvier (2015); I adopt a compromise age of 700 ± 100 Myr. For additional constraints on the behavior at early (< 100 Myr) and late (> 1 Gyr) ages, I further include the rotational moments of the 80 Myr α Persei and the 2.5 Gyr NGC 6819

(Irwin & Bouvier 2009; Meibom et al. 2015).

To quantify the agreement between a given rotational track and the cluster data, I adopt a reduced χ2 metric, taking into account the age uncertainty of each cluster and the rotation errors on each cluster moment. For a given stellar model, I determined the closest approach of its rotational track to each error ellipse, in terms of the σ oﬀset. I then summed the square of the σ values resulting from each data point, excluding those points employed during the calibration process, and divided the square root of this sum by the total number of data points to obtain a reduced

217 χ2. When ﬁtting multiple rotation moments simultaneously, I required the “best-ﬁt” age determined for each cluster to be equal in the two panels.

This methodology contains the implicit assumption that open clusters form a perfect evolutionary sequence. That is, a young cluster like the Pleiades, given time, will evolve into the rotational distribution revealed by older clusters, like the Hyades and M67. This assumption may break down if there exists substantial variance in the initial conditions of open clusters, which could inﬂuence e.g. the fraction of rapid vs. slow rotators. Having made this assumption explicit, I will cautiously proceed, and return to the question in §6.6.2.

6.2.2. Lithium data

For the mixing calibration, I choose as a comparison sample the data compilation of

Sestito & Randich (2005). These authors collected photometry and Li equivalent widths in open clusters available prior to 2005, grouped the clusters into age bins, and uniformly reanalyzed the Teﬀ s and Li abundances of their members in order to study the evolution of Li with time. I collected their analyzed data from the

WEBDA database, and for each of their speciﬁed age bins (see Sestito & Randich

(2005), table 3), computed the mean abundance within the Teﬀ bin 5750 ± 100 K, for comparison with my solar mass models. I noted in Somers & Pinsonneault

(2014) that their reported members for the 8 Gyr old NGC 188 cluster were in fact

218 sub-giants, rendering them an invalid comparison for the young clusters; accordingly,

I have excluded them from my analysis.

6.2.3. Stellar models

I calculate stellar evolution models using the Yale Rotating Evolution Code (YREC).

I adopt a present day solar metal abundance of Z/X = 0.02292 (Grevesse & Sauval

1998), accounting for helium and metal diﬀusion (Thoul et al. 1994), and calibrate the hydrogen abundance and mixing length such that a 1.0M stellar model ⊙

33 1 achieves the present-day solar luminosity of 3.8418 × 10 erg s− and solar radius of

6.9591 × 1010 cm at the solar age of 4.568 Gyr. My models use nuclear cross-sections from Adelberger et al. (2011), model atmospheres from Castelli & Kurucz (2004), equation of state tables from OPAL 2006 (Rogers, Swenson & Iglesias 1996; Rogers

& Nayfonov 2002), high and low temperature opacities from Mendoza et al. (2007) and from Ferguson et al. (2005) respectively, and the electron screening treatment of

Salpeter (1954).

To model stellar winds, I use a Rossby-scaled version of the the wind law of Chaboyer, Demarque & Pinsonneault (1995), a modiﬁcation of the original prescription of Kawaler (1988). This formulation includes three free parameters which must be calibrated: the normalization FK , the saturation threshold ωsat, and a measure of the magnetic ﬁeld geometry N.

219 Angular momentum loss obeys the formula,

N 1 N/3 R − M − f ωω4N/3,  K µR ¶ µM ¶ crit  ⊙ ⊙  τCZ,⊙  ω>ωcrit  τCZ  dJ  =  (6.1) dt  N 1 N/3 4N/3 R − M − ωτCZ  fK ω ,  µR ¶ µM ¶ µτCZ, ¶  ⊙ ⊙ ⊙  τCZ,⊙  ω ≤ ωcrit  τCZ   where M and R are the stellar mass and radius, ω is the rotation frequency, and τCZ is the convective overturn timescale. As I am modeling the slow, median, and fast bounds of the open cluster distribution, I require three separate initial rotation conditions. I choose initial rotation periods and disk-locking lifetimes which reproduce the slow, median, and fast bounds of the 13 Myr h Persei open cluster

(for details, see Somers & Pinsonneault 2015b). The three free parameters in the wind law are calibrated such that the median rotating model reproduces the median rotation rate of M37 at 550 Myr and the solar rotation rate at 4.568 Gyr, and such that the fast rotating model matches the 90th percentile rotators in the Pleiades. In the case of hydro models I relax the M37 constraint, as I ﬁnd they are intrinsically unable to predict both M37 and the Sun with a single wind law (see §6.5.1).

Internal angular momentum transport and chemical diﬀusion obey the coupled diﬀerential equations,

I dω d I dω ρr2 = f ρr2 (D + D ) , (6.2) M dt ω dr µ M hydro 0 dr ¶

220 dX d dX ρr2 i = f f ρr2D i . (6.3) dt c ω dr µ hydro dr ¶

Here, I/M is the moment of inertia per unit mass, Dhydro is the sum of diﬀusion coeﬃcients of the various hydrodynamic transport processes mentioned in §6.1

(Pinsonneault et al. 1989; Pinsonneault, Deliyannis & Demarque 1992; Chaboyer,

Demarque & Pinsonneault 1995), fω is a scaling factor that sets the eﬃciency of this transport (fω = 1 in my models), and fc is a scaling factor that sets the eﬃciency of transport of chemical species Xi relative to fω, related to anisotropic turbulence

(Chaboyer & Zahn 1992).

These details follow previous work, but in this chapter I introduce a new feature of the evolution code, a constant diﬀusion coeﬃcient D0 (e.g. Denissenkov et al.

2010). This value is a free parameter of the models, and enhances the diffusion of angular momentum by the specified value at every location in the radiative zone, but not the convective zone. This diffusion influences only angular momentum, and does not transport chemicals, and thus represents a background source of non-hydrodynamic transport. Its influence on stellar rotation and mixing are explored in detail in the following section.

6.3. Rotation and mixing in stellar models

In this section, I introduce the generic behavior of my rotating evolutionary models.

First, I describe the rotational and mixing properties of “hydro” and “solid body”

221 models. These are referred to collectively as the limiting cases, as they represent, respectively, the weakest and strongest possible coupling of the angular momentum histories of adjacent layers in the interiors of stars. Then, I use the limiting cases as a framework for introducing hybrid models, and describing how angular momentum transport in stellar interiors inﬂuences surface rotation and internal mixing.

6.3.1. Rotation in limiting case models

In hydro models, internal angular momentum transport is driven by structural evolution (particularly on the pre-main sequence) and by a number of hydrodynamic instabilities (particularly on the main sequence; see Pinsonneault 1997), which generally depend on the local rotation rate (e.g. meridional circulation), and/or the local rotation gradient (e.g. shear instabilities). Hydro models consider only these instabilities, locally conserving angular momentum otherwise, and thus represent the weakest possible coupling of a stellar interior permitted by hydrodynamics. Note that in the presence of strong horizontal turbulence, hydrodynamic transport is an inherently diﬀusive process, so angular momentum will preferentially move from layers that are rapidly rotating to layers which are slowly rotating (Chaboyer & Zahn

1992; Zahn 1992). This will generally result in outward diﬀusion in evolutionary models, as the only angular momentum sink is the magnetized wind at the surface.

The rotational evolution of the surface and core of a 1M , hydro model are ⊙ shown by the dashed and dotted lines in Fig. 1. Signiﬁcant diﬀerential rotation has

222 developed during the pre-main sequence, following the emergence of a radiative core.

After entrance onto the main sequence (∼ 30 Myr), the surface begins to rapidly spin down while the core remains nearly constant in its rotation. By ∼ 100 Myr, the internal differential rotation is so strong that the hydrodynamic transport processes which scale with the rotation gradient become influential. Diffusion of angular momentum from the core to the envelope slows the spindown rate of the surface by partially counter-balancing the loss at the surface, and the radiative core begins to gradually slow down. The surface reaches the solar rotation rate at 4.6 Gyr, but the core is still rotating ∼ 20× faster, strongly inconsistent with the near-solid-body behavior exhibited by the Sun (e.g. Eff-Darwich et al. 2002; Couvidat et al. 2003).

By contrast, solid body models force each layer in the radiative core to rotate with the same period as the convective envelope. Thus, angular momentum redistribution can be thought of as extremely eﬃcient in such models, as any loss from the surface is instantly balanced by transport from the interior. The rotational evolution of both the surface and core of a 1M , solid body model is shown by ⊙ the solid black line in Fig. 1. During the ﬁrst tens of Myrs, the surface rotation gradually increases due to Hayashi contraction. The peak rotation rate achieved at the zero-age main sequence is higher for the solid body case, because the lost surface angular momentum is instantaneously replenished, compared to the hundreds of

Myrs required in hydro models. Once on the main sequence, the star spins down at

223 a linear rate due to magnetized stellar winds, ﬁnally arriving at the solar rotation rate at 4.6 Gyr with no internal diﬀerential rotation.

6.3.2. Rotation in hybrid models

I now introduce my hybrid models which, in addition to the angular momentum diffusion induced by hydrodynamics, drive additional diffusion, the strength of which is constant in time but a free parameter. As this additional transport operates over a set timescale, its influence will be minimal at early times, but increasingly important with age. Note that hydro and solid body models are in fact special cases of hybrid models, where D0 is set to zero (no additional transport beyond hydrodynamics) or infinity (instantaneous coupling), respectively.

The rotational evolution of hybrid models with three different choices of D0 are shown in the right panel of Fig. 11. These hybrid models mimic hydro models in their surface rotation rates on the early main sequence, as insufficient time has passed for the constant diffusion coefficient to exert much influence. However, their core and surfaces converge in rotation rate over a much shorter time period due to the enhanced transport. This flux of angular momentum from the core to surface initially boosts the surface rotation rate relative to the hydro case around ∼ 50 Myr,

1 5 4 4 2 −1 The three choices of D0 represented in this plot, 1.5 × 10 , 8 × 10 , and 2 × 10 cm s , respectively correspond to approximate core-envelope coupling timescales of 15, 25, and 85 Myr for a 1M⊙ star.

224 and ultimately leads to solid-body-like behavior after a few hundred Myrs. The age at which this transition occurs depends on D0, with a stronger ﬂux leading to more rapid core-envelope coupling. The strength of internal diﬀerential rotation at the solar age is also a strong function of D0, but can reach essentially solid-body

4 2 1 behavior for D0 ∼> 5 × 10 cm s− (§6.6.1).

The basic picture that emerges from these considerations are that hybrid models resemble hydrodynamic models when young (highly decoupled), and resemble solid body models when old (highly coupled). This will prove important when I consider the mixing signal arising from the three transport paradigms.

6.3.3. Mixing

Mixing describes the transportation of chemical material between layers in the interior of stars. Hydrodynamic angular momentum transport is the vector for rotationally-driven mixing, and thus its history strongly inﬂuences the mixing pattern. Mixing has many observational tracers, but the most powerful is the surface abundance of lithium, which decreases when deep depleted material is mixed into the convective envelope, at a rate proportional to the mixing eﬃciency.

I set the initial Li abundance of the models to the primordial value of the Sun

(A(Li)2 ∼ 3.3), and present the evolution of the surface abundance predicted by the hydro and solid body models of §6.3.1 in the left panel of Fig. 6.2. These are

2A(Li) ≡ log(Li/H) + 12

225 compared to a standard stellar model which does not rotationally mix, shown by the thin grey line. The efficiency of mixing relative to angular momentum transport, a free parameter of the model (fc), is fixed to reproduce the solar abundance in the hydro model. For the purposes of illustrating the impact of D0 on the relative mixing strengths predicted by each transport scenario, fc is held fixed in this figure.

The behavior diﬀers in calibrated models (§6.5.2).

Each model undergoes a similar period of pre-main sequence depletion from

∼ 2−10 Myr, when the surface convection zone is deep. After this period, the rate of

Li depletion differs strongly between the two cases. The lack of significant differential rotation in the solid body model leads to very little hydrodynamic transport, and thus a very weak dilution of the surface Li abundances during the lifetime of the

Sun. By contrast, the hydro model develops extremely strong internal shears (Fig.

6.1), and drives abundant transport though hydrodynamics, thus leaving a far larger imprint on the surface Li abundance. Somewhat paradoxically, stronger coupling leads to less efficient mixing, as the strong differential rotation which so efficiently mixes chemical material is never allowed to develop in solid body models.

The right panel of Fig. 2 now shows the mixing signal derived from the three hybrid models of §6.3.2. Analogous to the rotation rates, the mixing signal of hybrid models follows a similar trajectory during the early years as do the hydro models

(up to ∼ 100 Myr). Thereafter, the internal shears become progressively suppressed by the enhanced angular momentum ﬂow, reducing the eﬃciency of hydrodynamic

226 transport and mixing. This leads to a gradual ﬂattening of the Li depletion track.

Thus, hybrid models are efficient mixers during their early years, and inefficient mixers at late ages. As before, the age of this transition is a strong function of D0, such that stronger transport leads to an early transition from efficient to inefficient, and thus a higher ultimate Li abundance.

I note that this late-time flattening is a generic prediction of models which are strongly differentially rotating on the early main sequence, and recouple over time, as it reflects a steady suppression of the shears in the interiors. Thus, even if my constant diffusion coefficient methodology does not capture the minutiae of internal angular momentum transport, the suppression of Li depletion at late ages should occur in any models which attempt to reproduce the flat solar rotation profile.

6.4. My calibration procedure

With the generic behavior of hybrid models laid out in §6.3, I now turn to my procedure for comparing these models to main sequence data. In §6.1, I briefly introduced the basic challenges, which include complications arising from accretion disk interactions, the pre-main sequence structural effects of magnetic fields, and uncertainties inherent to the solar calibration for both rotation and mixing. In this section, I detail my procedure for addressing these concerns.

First, during their formative years, protostars are surrounded by accretion disks. Proto-stars rotate well below break-up, indicating that angular momentum

227 is eﬃciently extracted by interactions with these disks (Vogel & Kuhi 1981; Shu et al. 1994). These stars would be expected to spin up dramatically as they contract, but there is a large population of slowly rotating stars, requiring a mechanism to spin down some systems early, but not all. Star-disk interactions are a natural explanation (Koenigl 1991). A rotation distribution initialized during this

“disk-locking” phase presents a poor initial launch point for models of wind-driven angular momentum loss. To circumvent this issue, I initialize the rotation rates on the 13 Myr old h Persei cluster (§6.2.1). This cluster is older than the canonical disk-lifetime upper limit of 6–10 Myr (e.g. Fedele et al. 2010). Thereafter, one can be assured that magnetized winds are the dominant mechanism robbing the star of angular momentum.

Second, we must account for the strong discrepancy between rotational mixing models on the pre-main sequence, and the empirical pattern. The same structural eﬀects that permit Li burning in the surface convection zone during the pre-main sequence make models highly sensitive to early rotationally-induced mixing. For this reason, mixing models tend to predict stronger pre-main sequence Li depletion with faster rotation (e.g. Pinsonneault et al. 1989; Charbonnel & Talon 2005; Somers &

Pinsonneault 2015b). While this correlation seems intuitive, the Li patterns of young

(∼< 150 Myr) open clusters indicate essentially no correlation between rotation and Li

228 depletion for solar-mass stars3. I illustrate this discrepancy with 1M hybrid models ⊙ in the left panel of Fig. 6.3. The range in initial rotation rates leads to a large dispersion by 125 Myr, in sharp contrast with the solar-temperature Li dispersion found in the ∼ 125 Myr (Stauﬀer et al. 1998) Pleiades and Blanco 1 open clusters

(black error bar). Evidently, rotation has a smaller impact on pre-main sequence Li depletion in 1M stars than predicted by mixing models. ⊙

In my recent papers (Somers & Pinsonneault 2015a,b), I suggested that this effect is a consequence of a correlation between rotation rate and radius inflation, likely driven by the enhanced magnetic activity and spot coverage of the fastest spinning cluster members. As a result, the rapid rotators are inflated during the pre-main sequence, cooling their interior and suppressing the rate of

Li depletion, both through direct destruction in the convection zone and through rotationally-induced depletion. Consequently, the fast and slow rotators have similar abundances at the zero-age main sequence. This suggestion is corroborated by the

Li-rotation correlation present in cooler stars (Soderblom et al. 1993a; Bouvier et al.

2016), which is likely produced through structural changes to the fastest rotators

(Somers & Pinsonneault 2015b).

Fortunately, as stars approach the main sequence, their convection zone retreats and Li depletion becomes less sensitive to radius inﬂation. Thus, Li depletion can

3A strong correlation does set in towards lower masses, but the sign is opposite of the rotational mixing prediction: faster rotating stars are more Li-rich than slow rotators!

229 proceed as predicted by mixing models thereafter. As I am concerned in this chapter solely with Li depletion arising on the main sequence, I can simply remove the complications inﬂuencing the early depletion signal by adopting a main sequence starting point. I therefore reset the Li abundances of the models to the median Li abundance of the Pleiades/Blanco 1 data set at 125 Myr, and track the evolution of Li thereafter. This technique is illustrated in the right panel of Fig. 6.3. In practice I ﬁnd little sensitivity to the precise age adopted, due to my freedom to calibrate the mixing strength, as long as it is in the range of 100–200 Myr. With the models initialized on the main sequence, I have avoided the complicated physics of the pre-main sequence, and can safely trace the depletion and dispersion developing thereafter.

Finally, the traditional calibration procedure for mixing calculations involved reproducing the solar rotation rate and the solar Li abundance with a median rotator by tweaking the free parameters controlling wind loss and internal chemical mixing.

While a useful strategy, two strong disadvantages of this approach are the lack of constraints on stellar behavior at earlier epochs, and the possibility that the Sun is atypical in its rotation and Li abundance. After all, there is no reason the Sun has to be a median star in a broad distribution of initial conditions. To address these concerns for the rotational calibration, I use the 550 Myr M37 as a joint calibration point alongside the Sun (§6.2.1). For the mixing calibration, I compute the average abundance of a series of open clusters of approximate age ∼ 600 Myr (the Hyades,

230 Praesepe, Coma Ber, and NGC 6633; Sestito & Randich (2005)), and calibrate mixing to a median age anchor. This has the advantage of marginalizing over many stars with the goal of ﬁnding a more stable median, and of making the late-age mixing a pure prediction of the physics. I will show in §6.5.2 that the evolution of Li abundances at ages ∼> 2 Gyr provide an excellent test of core-envelope recoupling, so an earlier calibration point is beneﬁcial.

These steps, alongside the hybrid models, constitute the theoretical and methodological advances presented in this chapter. In the subsequent section, I apply these principles to understand the angular momentum evolution and mixing history of solar-type stars.

6.5. Results

6.5.1. Constraining D0 with rotation data

Having introduced the generic behavior of hybrid models, I next appeal to open cluster data to discriminate between the three transport paradigms presented above.

I will show that hybrid models provide a superior description of the open cluster rotation pattern, and derive a best-ﬁt value for D0. This exercise will solidify a nominal rate of angular momentum transport for solar mass stars, thus allowing us to examine mixing in the next section.

231 Limiting Cases

Fig. 6.4 compares the evolutionary models with the rotation moments of Gallet &

Bouvier (2015) (see §6.2.1), represented by red ellipses. Considering ﬁrst the solid body models in solid grey, we ﬁnd good agreement in the slow (left) and median

(center) panels between the older clusters (∼> 500 Myr) and the solar-calibrated rotation track. However, the solid body models cannot spin down enough at early times to reproduce the slow and median rotators in the young clusters (∼< 500 Myr).

Turning to the hydro models (dashed grey), we find that the solar-calibrated track spins down far too much to reproduce the slow and median bounds. Furthermore, hydro models cannot reproduce either the slope or normalization of the spindown rate implied by the older clusters. Suggestively, the young cluster data appears in between these two limiting cases. Considering finally the most rapid rotators, we see that both the solid body and hydro models trace similar paths. This is likely because hydrodynamic transport scales very strongly with absolute rotation rate, so that in the fastest case, sufficient angular momentum is diffused even in hydro models to counteract the winds at the surface. Both cases predict reasonably well the evolution of the upper envelope of rotation rates, with the exception of M34 (see below). Despite this, both the solid body and hydro models fail spectacularly to

2 match the data (χred = 32 and 407, respectively), owing to their failures in the two slower rotation bins.

232 Hybrid Models

I next compare the hybrid models to the data. To do this, I varied the value of D0

2 until the best-ﬁt χred was obtained for the median rotator bin. The resulting best-ﬁt

4 2 1 D0 = 9 × 10 cm s− . Hybrid models with this value appear in orange in Fig. 6.4.

These models provide a far better qualitative agreement with the data, as their early (but not complete) decoupling reproduces the initial spin-down of the slow and median rotators younger than ∼ 500 Myr, and their steady angular momentum

ﬂux gradually drives the model towards the solid body track, thus matching the data beyond 500 Myr. The conclusion from these panels is clear: hybrid models signiﬁcantly outperform the adopted limiting cases, and demonstrate the need for core-envelope decoupling and recoupling during the main sequence.

2 The goodness-of-ﬁt for the median bin is an excellent χred = 1.04, demonstrating

2 quantitatively the superiority of hybrid models. However, we ﬁnd the χred in the slow and fast bins to be 3.99 and 2.04, respectively. I identify two factors which may

2 contribute to the poorness-of-fit. First, my χred minimization exercise did not allow for a rotation dependence in D0. To check this, I determined the best fit D0 for the slow and fast bins independently. We find that for the slow rotators alone, the

4 2 1 2 best-ﬁt D0 of 6.5 × 10 cm s− produces the improved χred = 2.88, and for the fast

5 2 1 2 rotators alone, D0 of 1.3 × 10 cm s− gives χred = 1.47. These results hint that a rotation dependence in the rate of the additional angular momentum transport

233 may be present. However, I caution that statistical noise from the next eﬀect may be responsible.

Second, the rotation moments from some clusters in the Gallet & Bouvier

(2015) sample are statistically inconsistent with one another. The most notable is

M50, whose slow rotation moment of 3.38 ± 0.30 ω is quite discrepant from the ⊙ equal-age Pleiades at 4.92 ± 0.13 ω . This disagreement is likely a consequence of ⊙ the high level of background-contamination in the M50 sample, estimated by the authors at 47% (Irwin et al. 2009). As the majority of background contaminants will be old, and thus slowly rotating, we would expect an artiﬁcially low value for the 25th percentile. Excluding M50 brings the best-ﬁt slow rotating model to

2 χred = 1.62, a signiﬁcant improvement. Furthermore, we cannot exclude similar contamination in the other data sets, or true variance in the rotation distribution between open clusters, which would further disrupt comparisons between my models and the data (e.g. Coker et al. 2016). For the fast rotating bin, a similar role is played by the anomalously rapid 90th percentile in M34. This could result from the inﬂuence of synchronized binaries, a more dominant rapid rotator population in this particular cluster (see §6.6.2), or may simply be shot noise. Excluding this datum

2 5 2 1 produces χred = 1.00 when D0 = 1.3 × 10 cm s− is adopted.

To summarize, while the limiting case models are strongly ruled out, hybrid models are able to reproduce the qualitative and quantitative evolution of the slow, median, and fast rotating moments of open clusters extremely well. In the next

234 section, I examine the mixing signal arising from the median rotating bin, and compare my results to time-series Li data.

6.5.2. Lithium depletion

I have shown that the median rotation distribution of solar-type stars can be well predicted if hydrodynamic angular momentum diﬀusion is supplemented by an

4 2 1 additional ﬂux of 9 × 10 cm s− . I now consider the mixing signal arising from such models, and compare their results to cluster Li data from Sestito & Randich (2005)

(see §6.2.2) as a function of age. I fix the mixing efficiency fc such that the median age Li bin containing the Hyades, Praesepe, Coma Ber, and NGC 6633 is reproduced, and examine the predictions for other age bins. I further apply my calibrated mixing efficiency to other Teff bins, in order to search for a mass dependence in the rate of core-envelope recoupling.

Solar Mass Results

Fig. 6.5 presents solar mass results. The black squares denote the mean Li abundance of the Teff = 5750 ± 100 K Li bin as a function of age, and their error bars indicate the standard deviation of the scatter. The red rectangle at 5 Gyr represents the median and standard deviation of the top half of the M67 data. The average abundance of the open clusters remain approximately flat until ∼ 200 Myr, when they begin to gradually decrease. By 2 Gyr, the average star has depleted ∼ 0.7 dex of Li. The final age bin, which consists solely of M67, is both substantially more

235 depleted, and contains a far larger scatter, compared to the pattern suggested by the younger clusters. I discuss possible reasons for this in §6.6.2, but note that the top half of the M67 distribution (red rectangle) coincides well with the depletion trend of the younger clusters.

The three lines show Li depletion tracks for solar-mass models calculated for the two limiting cases, and a hybrid model computed with the best-ﬁt, median-rotating,

4 2 1 solar-mass D0 of 9 × 10 cm s− . The mixing efficiency fc has been set for each model so that the Li abundance of the 600 Myr data point is accurately predicted; resulting values are 0.08, 0.21, and 9.50 for the hydro, hybrid, and solid body models, respectively. The difference between this result and Fig. 6.2 reflects the impact of calibrating the model Li depletion to a benchmark system. The limiting cases are shown simply for comparison, as their incompatibility with the rotation evolution of open clusters has already been demonstrated in §6.5.1. The hybrid model, on the other hand, was favored by §6.5.1, and agrees well with the shape of the trend up to 2 Gyr. It cannot account for the sudden drop off of the mean of M67, but does coincide with the upper envelope of the M67 distribution.

The visual success of the hybrid predictions for Li depletion in Fig. 6.5 are driven entirely by calibrating the free parameter fc, and are therefore not a compelling argument on their own. However, previous authors such as Denissenkov et al. (2010) and Lanzafame & Spada (2015) have found that the timescale for core-envelope coupling decreases with increasing mass. A ﬁxed mixing eﬃciency

236 therefore allows us to search for a mass dependence in the coupling rate implied by mixing. Thus, we expect that a ﬁxed D0 with mass produces poor agreement with the data, but an increasing D0 with mass will provide a better description.

Mass dependence of D0

With fc fixed by the solar Teff Li data, I can now make predictions for other masses. Namely, I compare the models to the average abundances of the three Teff bins considered by Sestito & Randich (2005): 6200 ± 150 K, 5900 ± 150 K, and

5600±100 K. I approximate these temperature bins respectively with models of mass

M = 1.15M , M = 1.05M , and M = 0.95M . Adopting the same treatment for ⊙ ⊙ ⊙ the pre-main sequence as in the solar mass case, I compute evolutionary models at these masses and compare them to the data in Fig. 6.6. Constant oﬀsets of −1 and

−2 dex have been applied to the 5900 K and 5600 K bins respectively. The models show reasonable agreement with the Li evolution for 5900 K and 5600 K (orange and blue), though compared to the mean values they are somewhat over-depleted and under-depleted, respectively. However, for the highest mass bin (green), the agreement is extremely poor. The models substantially over-deplete compared to the data at all ages, strongly ruling out moderately coupled models as an accurate description of 1.15M stars. Once again, M67 appears to defy the trend established ⊙ by the younger clusters: its mean is signiﬁcantly below the implied trend, its dispersion is far larger than any younger cluster, and its top-half distribution (black rectangles) appears to follow the trend implied by the other data.

237 I then allowed the constant diffusion coefficient, and thus the core-envelope coupling timescale, to vary for each mass individually, and solved for the D0 which reproduce the 600 Myr Li bin. These models appear in Fig. 6.7. Each colored line shows the preferred D0 for each mass bin, and the grey lines show the respective models from the previous figure. The best-fit D0 values reveal a strong mass trend in the required strength of the additional angular momentum transport mechanism, with increasingly large transport strengths required to reproduce the mixing signatures of higher mass stars (see the inset key). The preferred models agree reasonably well with the upper half of the M67 data in each bin, in accordance with the conclusion from the solar mass bin. In this context, variable core-envelope recoupling rates provides a natural explanation for the different magnitude of main sequence Li depletion observed in different temperature bins.

6.5.3. The timescales of core-envelope recoupling

In §6.5.2, I derived the constant diﬀusion coeﬃcients D0 that best reproduced the main sequence Li depletion signal for several stellar masses. In this section,

I calculate the core-envelope coupling timescales τCE corresponding to each D0, and compare the results to previous work. Note that this correspondence is only approximate, as the physics used to determine angular momentum transport diﬀers between the two-zone and hybrid models. To obtain these estimates, I use the

238 following equations from Denissenkov et al. (2010).

dJc (∆J)max 2 2 ˙ = − + rbceΩeMbce, (6.4) dt τCE 3

dJe (∆J)max 2 2 ˙ ˙ = − rbceΩeMbce + Jtot, (6.5) dt τCE 3 where Je and Jc represent the total angular momentum of the envelope and core, rbce represents the depth of the convection zone, Ωe is the rotation rate of the envelope,

M˙ bce is the rate of growth of the radiative core, and (∆J)max is set by,

IcIe (∆J)max = (Ωc − Ωe), (6.6) Ic + Ie

where I stands for moment of inertia. Each of these variables, except for τCE, is an output of my evolutionary models. I extract these values from the model runs, and determine the average τCE during the main sequence for a variety of masses and constant diffusion coefficients D0. The results appear in Fig. 6.8. The grey lines show curves of constant D0, and the core-envelope coupling timescale they produce as a function of mass. The timescales tend to fall off towards higher mass, due to the declining angular momentum content of the convective envelope as stars approach the Kraft break. The black X’s show the τCE values corresponding to the best-fit D0 for each mass bin from §6.5.2. From these points, we see that the Li data indicates quicker core-envelope recoupling in more massive stars. The mass trend is quite strong in this range, with the favored τCE’s for 0.95, 1.05, and 1.15M equal ⊙ to ∼ 32, 16, and 4 Myr, respectively.

239 We compare these results to the ﬁndings of Lanzafame & Spada (2015). They used rotation rates from open clusters to derive τCE, so my mixing-derived coupling timescales are an interesting comparison for theirs. Their results appear as the colored dots4. Each color represents diﬀerent modeling choices (see their table 3),

7.28 and the black line represents their power-law ﬁt to the data (τCE ∝ M − ). My results agree remarkably well, coinciding closely with the majority of the colored points from 0.95–1.05M . They did not consider 1.15M , but extrapolating from ⊙ ⊙ their power-law ﬁt and assuming similar sized error bars as their 1.10M value shows ⊙

9.1 1.8 probable ∼ 1σ agreement. A power-law fit to my points finds τCE ∝ M − ± , somewhat steeper than the value reported by Lanzafame & Spada (2015), but driven entirely by the highest mass point for which they report no data. All in all, we find good agreement with rotation-derived core-envelope coupling timescales, confirming

Li depletion as a powerful alternative constraint on internal angular momentum transport.

My measurements indicate an extremely rapid fall oﬀ in τCE above 1.1M . This ⊙ is unlikely to be an artifact of my main sequence calibration procedure: a calibration age earlier than 125 Myr would only serve to increase the required strength of τCE.

A later calibration age would decrease the required strength, but is unlikely as stellar activity, the presumptive mechanism which my calibration corrects for, has little

4 The dots have been shifted in mass slightly for increased visibility, but are measured at 0.05M⊙ intervals

240 eﬀect on rotational mixing. The error bars on my Li calibration point at 600 Myr are

5 2 1 rather large, but the weaker coupling value of ∼ 2 × 10 cm s− , required to match the Lanzafame & Spada (2015) extrapolation, produces too much Li depletion by

250 Myr to accommodate the earlier data points. A possible physical explanation explanation is the rapid disappearance of the convection zone as stars approach the

Kraft break, coupled with the increasing strength of gravity waves towards higher masses (e.g. Talon & Charbonnel 2003). The ﬁrst eﬀect substantially culls the amount of angular momentum contained in the envelope, and the second increasing the rate of angular momentum transport, so that recoupling occurs quite quickly.

However, I note that my adopted wind law begins to break down around this mass

(e.g. van Saders & Pinsonneault 2013), so deﬁnite quantitative results should be taken with caution. This result could be tested by measuring the core-envelope coupling in >1.15M stars through spot modulation studies. ∼ ⊙

While a mass-dependent background source of angular momentum flux, along with hydrodynamics, can reproduce well the Teff -dependent Li depletion pattern of open clusters, the source of this transport remains unknown. As mentioned in §6.1, gravity waves are a strong candidate, as they efficiently extract angular momentum from the deep core without directly mixing stellar material. Numerical calculations

ﬁnd timescales of a few 107 years for recoupling in solar mass stars (Zahn et al.

1997; Kumar & Quataert 1997), and a increasing momentum ﬂux towards higher masses, up to the Kraft break (Talon & Charbonnel 2003), consistent with my

241 derived normalization and trend. By contrast, magnetic ﬁelds do not predict a mass dependence in the rate of recoupling (e.g. Charbonneau & MacGregor 1993; Barnes et al. 1999), in disagreement with my conclusions.

6.6. Discussion

6.6.1. The solar rotation proﬁle

An additional empirical constraint on the efficiency of core-envelope coupling is the internal rotation profile of the Sun. Numerous authors have used helioseismic inversions to infer the solar rotation profile, and have found it to be extraordinarily

flat within the radiative core, down to ∼ 0.3R (for an extensive review, see Howe ⊙ 2009). These findings strongly contradict the predictions of “hydro” models, which retain substantial internal differential rotation at the solar age (see §6.3.1). However, hybrid models, due to their enhanced transport, dramatically suppress the degree of late-age differential rotation. While such a simplified model cannot be expected to reproduce all the details of the solar rotation profile, it can provide reasonable limits on the average rate of angular momentum needed to supplement hydrodynamic transport in the Sun.

Fig. 6.9 shows the internal rotation proﬁles extracted for a range of D0

4 2 1 values. From top to bottom, the strength of D0 increases from 3 × 10 cm s− to

5 2 1 1.5 × 10 cm s− , with my preferred solar case highlighted in red. The convective

242 region, which extends down to ∼ 0.72R , is solid body by design, but below the ⊙ tachocline we see the onset of radial diﬀerential rotation. Diﬀerential rotation below the surface convection zone is measured to be of order 10% in the Sun (e.g.

Couvidat et al. 2003), which I take as a reasonable overall bound on diﬀerential

4 2 1 rotation in my models. This is achieved for D0 ∼> 5 × 10 cm s− , corresponding to a core-envelope coupling timescales τCE ∼ 35 Myr; this represents the weakest possible average rate of transport required to match the Sun. My best-ﬁt solar value

4 2 1 of D0 ∼ 9 × 10 cm s− produces coupling above this level, showing that the derived rates of angular momentum transport are consistent with the Sun.

6.6.2. Why is M67 so diﬀerent?

I noted in §6.5 that the oldest cluster in my Li sample, M67, seemed to deviate from the trend established by the young clusters. Namely, its average abundance shows a steeper decline from the clusters at 2 Gyr compared to the prior trend, and its dispersion is larger by 1.5-4× compared to the other points, depending on the temperature bin. In this section, I speculate on the cause of this diﬀerence, focusing in particular on how to reconcile the 2 Gyr old clusters with the 4 Gyr M67. I see three possible explanations: 1) incompleteness of the Li samples, 2) additional mixing setting in after ∼ 2 Gyr, 3) diﬀerences in the initial rotation distributions of open clusters.

243 First, it is possible that the 2 Gyr clusters are as Li-dispersed as M67, but observations have failed to capture this dispersion due to small number statistics.

This explanation has been previously explored by Sestito et al. (2004), who computed the likelihood of the relatively narrow Teff -Li relation of NGC 752 (2 Gyr) and the highly dispersed Teff -Li relation of M67 (4 Gyr) being drawn from the same distribution at two different ages. They demonstrated at the 4σ level that NGC 752 does not possess as many stars depleted in Li by >5× as M67, thus concluding that the dispersions of the two clusters are intrinsically different. Furthermore, the three

Li-studied 2 Gyr clusters (NGC 752, IC 4651 and NGC 3680) show no statistical evidence for signiﬁcant diﬀerences in their Li patterns (Sestito et al. 2004; Somers &

Pinsonneault 2014), pinpointing M67 as the outlier. This leaves two explanations: either the dispersion appears after 2 Gyr, or sets in before 2 Gyr in some, but not all, open clusters.

If one believes that the open clusters form a true evolutionary sequence (an assumption I have made in this chapter), then in order to explain M67, a Li depletion mechanism must kick after a few Gyr which is not only stronger than the depletion occurring on the early main sequence, but is highly variable between stars. Most depletionary processes are insufficiently strong and do not vary from star to star at the required level, such as mass loss (Swenson & Faulkner 1992) and microscopic diffusion (Richer & Michaud 1993). On the other hand, rotational mixing can in principle vary between stars with different rotations rates. However,

244 rotational mixing tends to decrease in eﬃciency with advancing age, due to the gradual spindown of stars. Moreover, the converging rotation rates of the core and envelope suppress internal shears, further inhibiting mixing. Finally, the spread in rotation rates necessary to drive unequal Li depletion in diﬀerent cluster members is only present in young (∼< 500 Myr) clusters, and is certainly gone by 2.5 Gyr

(Meibom et al. 2015), let alone the age of M67 (e.g. Barnes et al. 2016), due to the progressive convergence of rotation rates at ﬁxed mass. Given the patent failure of the best candidate mechanism, the existence of a physical process which created the

M 67 Li spread after 2 Gyr would be quite surprising, and would clearly represent new physics. While we cannot rule out this possibility, I consider it an unlikely explanation.

The only remaining possibility that occurs to us is that M67 had an intrinsically diﬀerent mixing history than the other clusters in the sample. For instance, if M67 was born with far more rapid rotators than the other clusters, a larger spread would be a natural prediction of the rotational mixing framework. Inhomogeneity in initial rotation distributions has been recently demonstrated by Coker et al. (2016), who found that the 13 Myr h Persei is rich in rapid rotators compared to the 1 Myr

Orion Nebula Cluster. The cause is unclear, but may relate to the number density of proto-stars in the early cluster environment, and the subsequent inﬂuence of dynamical interactions between cluster members and the circum-stellar disks of their siblings. Such interactions could truncate the disk lifetime, thus terminating

245 its extraction of angular momentum from its host star, resulting in greater rotation rates. This bounty of rapid rotators then undergo severe Li depletion on the early main sequence, creating a spread which persists for the lifetime of the cluster.

To illustrate this possibility, I generated synthetic open clusters with diﬀerent initial rotation distributions. First, I extracted rotation rates from h Persei members between 0.9 and 1.1M from the catalog of Moraux et al. (2013). Their rotation ⊙ distribution appears as a black histogram in Fig. 6.10. This distribution is clearly double peaked, which I identify as rapid and slow rotating components. I ﬁt the distribution with a double Gaussian to determine the peak and width of the two components. I then varied the normalization of the two peaks in order to generate a rapid-rotator-poor and rapid-rotator-rich cluster distributions, with, respectively,

5% and 30% of members drawn from the fast component. Finally, I forward modeled these synthetic clusters to the main sequence, normalizing on the Pleiades abundance as before, and compared the predictions to stars between 0.95–1.05M in the Li ⊙ catalog of M67 (Pace et al. 2012). For comparison, I show two additional open clusters, whose abundances have been analyzed by Castro et al. (2016) in a fashion consistent with these M67 data: the Hyades and NGC 752.

The comparison appears in Fig. 6.11. The cluster data appear in blue, with detections indicated with circles and upper limits with triangles. One immediately notices the striking dispersion in M67, accentuated by the large number of upper limits. The grey histograms alongside each cluster represent the number of expected

246 members, as a function of A(Li), for the rapid-rotator-poor distribution on the left, and the rapid-rotator-rich distribution on the right. For the left panel, the main peak is strongly focused around the higher abundance stars, corresponding to the location of the totality of the members of the Hyades and NGC 752, with the possible exception of the lone upper limit at 2 Gyr. The red points represent a single sampling of the distribution, revealing a tight clustering at the high end, with a few rapid rotators which have been heavily depleted. This distribution is a good match to the two younger clusters, but is strongly inconsistent with M67, with its ample spread and numerous upper limits. By contrast, the rapid-rotator-rich histogram in the right panel is double-peaked, permitting a large number of Li-rich and Li-poor members. The random sampling shows a far more even sprinkling down to very low abundances, due to its strong rapid rotator component. While clearly inconsistent with the younger two clusters, the large number of heavily depleted stars presents a reasonable facsimile of M67.

While merely a illustrative case with a simplistic initialization, this exercise makes clear the extent to which initial rotation conditions inﬂuence the subsequent rotational mixing pattern in a given open cluster. It also makes a salient prediction: large Li dispersions caused by rotational mixing must set in early, when stars still spin rapidly. Thus, if this scenario holds, not all clusters of similar age and metallicity should host equivalent Li patterns. If the number of rapid rotators in the cluster is indeed related to the density of the star-forming environment, then a

247 correlation between cluster density and Li spread should be latent. An attractive target for testing this theory is the dense open cluster M37, whose rotation pattern has been well studied but who lacks any Li measurements to date. A larger Li dispersion in this cluster compared the the similar-age Hyades and Praesepe would be powerful conﬁrmation that Li spreads are a fossil remnant of the rotational properties of stars in their earliest days.

6.7. Summary and Conclusions

In order to investigate the impact of core-envelope coupling on rotational mixing,

I have computed rotating stellar models whose angular momentum transport is supplemented by a background source of transport, parameterized by my models as a constant diﬀusion coeﬃcient (D0); I refer to these as “hybrid models”. To compare these to data, I devise a calibration methodology which accounts for several previously confounding aspects of the evolution of Li abundances in open clusters.

These include the impact of the early disk-locking phase on the angular momentum evolution of young stars, the shortcomings of using the Sun as the sole calibration point for rotational models, and the anomalous rates of Li destruction on the pre-main sequence, which I argue are suppressed by a cooling of the convection zone by strong magnetic activity. By employing a post-disk-locking cluster for choosing initial rotation rates, a main sequence initialization point for Li abundances, and an ensemble of open clusters to constrain the angular momentum evolution and mixing

248 eﬃciency, I present a modernized approach to studying Li abundances on the main sequence and beyond.

My hybrid models behave similar to models which employ only hydrodynamic transport at early ages, and similar to solid-body models at late ages. As a consequence, while the traditional limiting cases fail to predict both the early and late evolution of the distribution of rotation rates in open clusters, hybrid models provide an excellent match for the observed pattern across the entire main sequence.

Solar-mass models with a core-envelope coupling timescale ∼ 21 Myr provide a good description for the median evolution of the open cluster rotation pattern, proving that additional sources of transport are at work in solar-type, main-sequence stars.

I also ﬁnd a suggestion of a rotation dependence in the rate of this transport, such that faster stars drive stronger coupling.

Using open cluster Li abundances, I ﬁnd that the required strength of D0 is a strong function of mass, with stronger angular momentum transport required to explain the mixing patterns of higher mass stars. This implies a strong mass dependence in the rate of core-envelope recoupling, with timescales of ∼ 32, 16, and 4 Myr favored respectively for 0.95, 1.05, and 1.15M . This dependence, ⊙ derived entirely from mixing data, agrees well with the mass dependence revealed by rotation data. My results shows a similar mass dependence as that predicted by gravity wave angular momentum transport, and can account for the near-solid-body rotation proﬁle of the Sun. Finally, I show that M67 has a much lower median Li

249 abundance, and a far larger scatter, than any other open cluster I have considered. I suggest that this results from diﬀerences in the initial distribution of rotation rates in the formative years of the cluster, perhaps associated with the density of the star forming region. The dense open cluster M37 is a valuable test bed for this theory, were its Li properties to be studied.

These results, along with ﬁndings in my previous work (Somers & Pinsonneault

2015a,b), provide a complete story for the evolution of Li abundances in solar-type stars. During the pre-main sequence, Li depletion is reduced in rapidly-rotating objects by the profound effects of magnetic activity on the structure of stellar interiors. This suppresses the dispersion which would otherwise develop due to pre-main sequence rotational mixing. Once on the main sequence, the convection zone retreats and direct destruction of Li in the convection zones ceases. However, the strong shears which develop at the onset of the main sequence, and the progressive spindown of the surface due to magnetic torques, leads to a depletion of the surface abundance through rotational mixing. As the main sequence progresses, the core and envelope recouple on a timescale governed by both their mass and rotation rate, gradually reducing the rate of depletion, and plateauing to a fixed abundance at a late age. This picture makes a specific prediction about the Li abundances of old open clusters – their abundances will reflect a slowing of Li depletion at late ages.

250 10 10 ⊙ ⊙ ω ω / / / ω ω

D =1.5 ×105 cm2 s−1 Solid Body 0 1 1 4 2 −1 Hydro (surface) D0 =8.0 ×10 cm s 4 2 −1 Hydro (core) D0 =2.0 ×10 cm s 10 100 1000 10000 10 100 1000 10000 Age (Myr) Age (Myr)

Fig. 6.1.— Angular momentum evolution of limiting case and hybrid models. The surface and core rotational evolution of solar mass models, employing various internal angular momentum transport paradigms. The models include the surface and core of the solid-body model (solid black), the hydro model (dashed and dotted black), and hybrid models with three different values of the constant diffusion coefficient in blue, orange, and green. Each model’s wind law has been calibrated to reach the solar rotation rate at the solar age.

251 3.5 3.5

3.0 3.0

2.5 2.5

A(Li) 2.0 A(Li) 2.0

1.5 1.5

5 2 −1 D0 =1.5 ×10 cm s 1.0 1.0 4 2 −1 Solid Body D0 =8.0 ×10 cm s 4 2 −1 Hydro D0 =2.0 ×10 cm s 1 10 100 1000 10000 1 10 100 1000 10000 Age (Myr) Age (Myr)

Fig. 6.2.— Lithium depletion of limiting case and hybrid models. Surface Li depletion for the solid-body (solid black), hydro (dashed black), and hybrid (colored) models of Fig. 6.1. The mixing parameter fc has been calibrated such that the hydro model attains the Li abundance of the Sun at the solar age, and the other models are shown with the same mixing efficiency for comparison. Stronger core-envelope coupling leads to less efficient mixing at late ages, causing hybrid models to flatten their Li curves after several hundred Myrs. The thin grey line represents a non-rotating model.

252 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0

A(Li) 1.5 A(Li) 1.5

1.0 Slow 1.0 Slow Median Median 0.5 Fast 0.5 Fast 0.001 0.01 0.1 1 10 0.1 1 10 Age (Gyr) Age (Gyr)

Fig. 6.3.— Illustration of my calibration procedure. An illustration of my pre-main sequence detrending process. Left: The Li abundance tracks of three hybrid models of varying initial rotation rate. Li depletion is a strong function of rotation rate in the models, but this is contradicted by Li data from the Pleiades and Blanco 1 (black point), and explained by the structural eﬀects of starspots (Somers & Pinsonneault 2015a). Right: To account for this early failure, I reset the Li abundance of the tracks to the Pleiades abundance at 150 Myr, and track the depletion thereafter.

253 Solid Body Solid Body Solid Body Hydrodynamic Hydrodynamic 100 Hydrodynamic Hybrid Model Hybrid Model Hybrid Model Pleiades

10 10 Pleiades Pleiades ⊙ ⊙ ⊙

ω ω ω 10 / / / Praesepe Praesepe ω ω ω Praesepe Slow Median Fast Rotators Rotators Rotators

1 1 1 hPer AlphaPer M50 M35 M37 Hyades N6811 N6819 hPer AlphaPer hPer AlphaPer M50 M35 M37 Hyades N6811 N6819 M50 M35 M37 Hyades N6811 N6819 10 100 1000 10000 10 100 1000 10000 10 100 1000 10000 Age (Myr) Age (Myr) Age (Myr)

Fig. 6.4.— Constraining my models with open cluster rotation data. A comparison of the surface rotation rates of my calibrated solid body, hydrodynamic, and hybrid 4 2 1 models (calculated with D0 = 9 × 10 cm s− ) and the rotation moments of Gallet & Bouvier (2015), plotted as red 1σ error ellipses. Left: Solid body models do not spin down enough at early ages, and hydrodynamic models spin down too much at early ages, to explain the slowly rotating data. However, hybrid models spin down the right amount, and converge at later ages, producing better description of the data. Center: Similarly, hybrid models out-perform the two limiting cases for describing the median rotators. Right: Each rotation case performs reasonably similarly for the rapid rotators, owing to the strong hydrodynamic transport driven by rapid rotation.

254 Model mass = 1.00 M 3.0 ⊙ Teff = 5750 ±100K

2.5

2.0 A(Li)

1.5 Solid Body Hydro 4 2 −1 D0 =9.0 ×10 cm s 1.0 0.1 1 10 Age (Gyr)

Fig. 6.5.— Constraining my lithium depletion models with solar mass cluster data. Li abundances as a function of age (black squares) in the Teff range 5750 ± 100 K from Sestito & Randich (2005), compared to solar mass models computed with the three angular momentum transport paradigms. The mixing efficiency fc is calibrated for each model such that the data point at 600 Myr is reached. The two limiting cases predict a turn-down at late ages, while the hybrid models flatten. The final cluster, M67, appears to buck the prevailing trend, but the upper third of the data (red rectangle) matches the trend better.

255 3

1.15 M ⊙ 6200 ±150K Const. = 0

1.05 M ⊙ 5900 ±150K Const. = -1

1 A(Li) + Const. A(Li) +

0.95 M ⊙ 5600 ±100K 0 Const. = -2

-1 4 2 −1 D0 =9.0 ×10 cm s 0.1 1 10 Age (Gyr)

Fig. 6.6.— Solar-calibrated hybrid models in different mass bins. Li abundance as a function of age compared to fixed-D0 hybrid models in three different Teff bins: 6200 ± 150 K (green), 5900 ± 150 K (orange), and 5600 ± 100 K (blue), with constant offsets applied in order to display on a single plot. Black lines represent, from top to bottom, 1.15M , 1.05M , and 0.95M hybrid models, with the mixing efficiency fc ⊙ ⊙ ⊙ and constant diffusion coefficient D0 as determined by the solar mass calibration in §6.5.1 and 6.5.2. These models slightly under-predict Li depletion in the low mass bin, slightly over-predict Li depletion in the middle mass bin, and massively over-predict Li depletion in the high mass bin.

256 3

1.15 M ⊙ 6200 ±150K Const. = 0

1.05 M ⊙ 5900 ±150K Const. = -1

1 A(Li) + Const. A(Li) +

0.95 M ⊙ 5600 ±100K 0 Const. = -2

5 2 −1 D0 =4.7 ×10 cm s 5 2 −1 D0 =1.3 ×10 cm s 4 2 −1 D0 =9.0 ×10 cm s -1 4 2 −1 D0 =5.5 ×10 cm s 0.1 1 10 Age (Gyr)

Fig. 6.7.— The best fit hybrid models for each mass bin. Li abundance as a function of age compared to variable-D0 hybrid models, in the same three Teff bins as Fig. 4 2 1 6.6. Grey lines re-plot the D0 = 9 × 10 cm s− models from Fig. 6.6, and the green, orange, and blue lines plot the hybrid models with the value of D0 which best predicts the Li data. Each model uses the mixing efficiency fc which was fixed by the solar mass case. Higher mass stars require systematically stronger angular momentum transport in order to predict the observed Li pattern. This demonstrates a mass dependence in the rate of core-envelope recoupling.

257 300

1 ×104 cm2 s−1

100

2 ×104 cm2 s−1

30 5 ×104 cm2 s−1

1 ×105 cm2 s−1 10

2 ×105 cm2 s−1 Coupling Timescale (Myr) Timescale Coupling

3 4 ×105 cm2 s−1 0.8 0.9 1.0 1.1 1.2 Mass

Fig. 6.8.— My best fit core-envelope coupling timescales. The approximate timescales of core-envelope recoupling (τCE) corresponding to my best-fit hybrid models. Grey lines show curves of constant D0, and the τCE they induce as a function of mass. My derived values of D0 for 0.95M , 1.0M , 1.05M , and 1.15M appear as black ⊙ ⊙ ⊙ ⊙ X’s. My data are compared to the τCE values derived from open cluster rotation by Lanzafame & Spada (2015), represented by the colored dots. The black line is a power-law fit to their points. Agreement is overall excellent, confirming Li as a strong tracer of core-envelope recoupling.

258 1.20 3.0 ×104 Weak 4.0 ×104 Coupling 5.0 ×104 1.15 6.0 ×104 7.0 ×104 9.0 ×104 ⊙ 1.10 1.1 ×105 1.5 ×105 / 1.05

Strong 1.00 Coupling 0.0 0.2 0.4 0.6 0.8 1.0 R/R ⊙

Fig. 6.9.— Rotation profiles of solar-mass, solar-age hybrid models. The internal rotation profile of several solar-mass hybrid models at the age of the Sun. My 4 2 1 preferred D0 of 9 × 10 cm s− shows a 4% core-envelope differential rotation, within the observational limits derived for the Sun of ∼< 10%. This suggests my models predict strong enough angular momentum transport to explain the solar profile.

259 15

10 Number

0 0.1 1 10 Rotation Period

Fig. 6.10.— Double Gaussian fit of the h Persei rotation distribution. The black histogram shows rotation rates of 0.9–1.1M h Persei cluster members, as derived ⊙ by Moraux et al. (2013). The blue solid line represents a double-Gaussian fit to the double-peaked profile, and the over-laid red and green dashed lines show the individual Gaussians. The lower red dashed line, and the upper green dashed line, show an example of a re-normalized initial distribution with a lower rapid rotator fraction than h Per.

260 3 5% Rapid Rotators 3 30% Rapid Rotators

2 2

1 1 A(Li) A(Li)

0 0

-1 -1 0.1 1 10 0.1 1 10 Age (Gyr) Age (Gyr)

Fig. 6.11.— Forward models of clusters with different rapid rotator fractions. Forward modeled Li abundances for synthetic clusters with different initial rotation distributions. Left: Blue points represent 0.95-1.05M members of, from left to ⊙ right, the Hyades, NGC 752, and M 67, randomly offset in age for visibility. Circles reflect detections, and triangles represent upper limits. The grey histograms show the relative fractions of expected stars at different abundances for a population with 5% rapid rotators (red points are one such realization). Both the Hyades and NGC 752 are consistent with few rapid rotators, but M 67 is not. The lines reflect Li depletion histories for three different initial rotation rates. Right: Same as the left, but for a synthetic population with a 30% rapid rotator fraction. This distribution is a better fit to the wide dispersion and numerous upper limits in M 67, but strongly contradicts the two younger clusters. This may be a hint that the clusters were born with different rotation distributions.

261 References

Adelberger, E. G., Garc´ıa, A., Robertson, R. G. H., et al. 2011, Reviews of Modern Physics, 83, 195

Alexander, D. R., & Ferguson, J. W. 1994, ApJ, 437, 879

Allard, F., Hauschildt, P. H., Alexander, D. R., & Starrﬁeld, S. 1997, ARA&A, 35, 137

An, D., Terndrup, D. M., Pinsonneault, M. H., et al. 2007, ApJ, 655, 233

Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197

Ando, H. 1986, A&A, 163, 97

Andronov, N., & Pinsonneault, M. H. 2004, ApJ, 614, 326

Anthony-Twarog, B. J., Deliyannis, C. P., Twarog, B. A., Croxall, K. V., & Cummings, J. D. 2009, AJ, 138, 1171

Argiroﬃ, C., Caramazza, M., Micela, G., Moraux, E., & Bouvier, J. 2013, Protostars and Planets VI Posters, 89

Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481

Balachandran, S. 1995, ApJ, 446, 203

Balachandran, S. C., Mallik, S. V., & Lambert, D. L. 2011, MNRAS, 410, 2526

Baraﬀe, I., Chabrier, G., & Gallardo, J. 2009, ApJ, 702, L27

Baraﬀe, I., & Chabrier, G. 2010, A&A, 521, A44

Baraﬀe, I., Vorobyov, E., & Chabrier, G. 2012, ApJ, 756, 118

Barnes, S. A. 2007, ApJ, 669, 1167

Barnes, G., Charbonneau, P., & MacGregor, K. B. 1999, ApJ, 511, 466

262 Barnes, S. A., Weingrill, J., Fritzewski, D., & Strassmeier, K. G. 2016, ArXiv e-prints

Basri, G., Marcy, G. W., & Graham, J. R. 1996, ApJ, 458, 600

Bastien, F. A., Stassun, K. G., Basri, G., & Pepper, J. 2013, Nature, 500, 427

Bastien, F. A., Stassun, K. G., Pepper, J., et al. 2014, AJ, 147, 29

Bell, C. P. M., Naylor, T., Mayne, N. J., Jeﬀries, R. D., & Littlefair, S. P. 2013, MNRAS, 434, 806

Berdyugina, S. V. 2005, Living Reviews in Solar Physics, 2, 8

Berger, D. H., Gies, D. R., McAlister, H. A., et al. 2006, ApJ, 644, 475

Bildsten, L., Brown, E. F., Matzner, C. D., & Ushomirsky, G. 1997, ApJ, 482, 442

Binks, A. S., & Jeﬀries, R. D. 2014, MNRAS, 438, L11

Boesgaard, A. M., & Steigman, G. 1985, ARA&A, 23, 319

Boesgaard, A. M., & Tripicco, M. J. 1986, ApJ, 302, L49

Boesgaard, A. M., & Friel, E. D. 1990, ApJ, 351, 467

Boesgaard, A. M., Roper, B. W., & Lum, M. G. 2013, ApJ, 775, 58

B¨ohm-Vitense, E. 1958, ZAp, 46, 108

Bouvier, J. 1990, AJ, 99, 946

Bouvier, J., Rigaut, F., & Nadeau, D. 1997, A&A, 323, 139

Bouvier, J. 2008, A&A, 489, L53

Bouvier, J., Lanzafame, A., Venuti, L., et al. 2016, ArXiv e-prints

Boyajian, T. S., McAlister, H. A., Baines, E. K., et al. 2008, ApJ, 683, 424

Boyajian, T. S., von Braun, K., van Belle, G., et al. 2012, ApJ, 757, 112

Brandt, T. D., & Huang, C. X. 2015, ApJ, 807, 24

Burke, C. J., Pinsonneault, M. H., & Sills, A. 2004, ApJ, 604, 272

Cargile, P. A., James, D. J., & Jeﬀries, R. D. 2010, ApJ, 725, L111

Carlsson, M., Rutten, R. J., Bruls, J. H. M. J., & Shchukina, N. G. 1994, A&A, 288, 860

263 Casagrande, L., Flynn, C., Portinari, L., Girardi, L., & Jimenez, R. 2007, MNRAS, 382, 1516

Casagrande, L., Ram´ırez, I., Mel´endez, J., Bessell, M., & Asplund, M. 2010, A&A, 512, A54

Casagrande, L., Portinari, L., Glass, I. S., et al. 2014, MNRAS, 439, 2060

Castelli, F., & Kurucz, R. L. 2004, ArXiv Astrophysics e-prints

Castro, M., Do Nascimento, J. D., Jr., Biazzo, K., Mel´endez, J., & de Medeiros, J. R. 2011, A&A, 526, A17

Castro, M., Duarte, T., Pace, G., & Dias do Nascimento, Jr, J. 2016, ArXiv e-prints

Chaboyer, B., & Zahn, J.-P. 1992, A&A, 253, 173

Chaboyer, B., Demarque, P., & Pinsonneault, M. H. 1995, ApJ, 441, 865

Chaboyer, B., Demarque, P., & Pinsonneault, M. H. 1995, ApJ, 441, 876

Chabrier, G., Gallardo, J., & Baraﬀe, I. 2007, A&A, 472, L17

Charbonneau, P., & MacGregor, K. B. 1993, ApJ, 417, 762

Charbonnel, C., & Talon, S. 2005, Science, 309, 2189

Clausen, J. V., Bruntt, H., Claret, A., et al. 2009, A&A, 502, 253

Coker, C. T., Pinsonneault, M., & Terndrup, D. M. 2016, ArXiv e-prints

Cottaar, M., Covey, K. R., Meyer, M. R., et al. 2014, ApJ, 794, 125

Couvidat, S., Garc´ıa, R. A., Turck-Chi`eze, S., et al. 2003, ApJ, 597, L77

Cranmer, S. R., & Saar, S. H. 2011, ApJ, 741, 54

Cummings, J. 2011, Ph.D. Thesis,

Cummings, J. D., Deliyannis, C. P., Anthony-Twarog, B., Twarog, B., & Maderak, R. M. 2012, AJ, 144, 137

Cyburt, R. H., Fields, B. D., & Olive, K. A. 2004, Phys. Rev. D, 69, 123519

Cyburt, R. H., Fields, B. D., & Olive, K. A. 2008, JCAP, 11, 12

D’Antona, F., & Mazzitelli, I. 1994, ApJS, 90, 467

D’Antona, F., & Mazzitelli, I. 1997, Mem. Soc. Astron. Italiana, 68, 807

264 D’Orazi, V., & Randich, S. 2009, A&A, 501, 553

Dahm, S. E. 2008, Handbook of Star Forming Regions, Volume I, 966

Denissenkov, P. A., Pinsonneault, M., Terndrup, D. M., & Newsham, G. 2010, ApJ, 716, 1269

Dobbie, P. D., Lodieu, N., & Sharp, R. G. 2010, MNRAS, 409, 1002

Duncan, D. K., & Jones, B. F. 1983, ApJ, 271, 663

Eddington, A. S. 1929, MNRAS, 90, 54

Eﬀ-Darwich, A., Korzennik, S. G., & Jim´enez-Reyes, S. J. 2002, ApJ, 573, 857

Eggenberger, P., Maeder, A., & Meynet, G. 2005, A&A, 440, L9

Eggenberger, P., Haemmerl´e, L., Meynet, G., & Maeder, A. 2012, A&A, 539, A70

Endal, A. S., & Soﬁa, S. 1978, ApJ, 220, 279

Epstein, C. R., & Pinsonneault, M. H. 2014, ApJ, 780, 159

Fedele, D., van den Ancker, M. E., Henning, T., Jayawardhana, R., & Oliveira, J. M. 2010, A&A, 510, A72

Feiden, G. A., & Chaboyer, B. 2012, ApJ, 757, 42

Feiden, G. A., & Chaboyer, B. 2012, ApJ, 761, 30

Feiden, G. A., & Chaboyer, B. 2013, ApJ, 779, 183

Feiden, G. A., & Chaboyer, B. 2014, ApJ, 789, 53

Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005, ApJ, 623, 585

Fleming, T. A., Gioia, I. M., & Maccacaro, T. 1989, ApJ, 340, 1011

Ford, A., Jeﬀries, R. D., & Smalley, B. 2005, MNRAS, 364, 272

Fran¸cois, P., Pasquini, L., Biazzo, K., Bonifacio, P., & Palsa, R. 2013, A&A, 552, A136

Fricke, K. 1968, ZAp, 68, 317

Gallet, F., & Bouvier, J. 2013, A&A, 556, A36

—. 2015, A&A, 577, A98

Goldreich, P., & Schubert, G. 1967, ApJ, 150, 571

265 Gough, D. O., & McIntyre, M. E. 1998, Nature, 394, 755

Gough, D. O., & Tayler, R. J. 1966, MNRAS, 133, 85

Grevesse, N., & Sauval, A. J. 1998, Space Sci. Rev., 85, 161

Haisch, K. E., Jr., Lada, E. A., & Lada, C. J. 2001, ApJ, 553, L153

Hartman, J. D., Gaudi, B. S., Pinsonneault, M. H., et al. 2009, ApJ, 691, 342

Hartman, J. D., Bakos, G. A.,´ Kov´acs, G., & Noyes, R. W. 2010, MNRAS, 408, 475

Hartmann, L. 2001, AJ, 121, 1030

Hartmann, L., Cassen, P., & Kenyon, S. J. 1997, ApJ, 475, 770

Heiter, U., Soubiran, C., Netopil, M., & Paunzen, E. 2013, arXiv:1311.2306

Herbig, G. H. 1965, ApJ, 141, 588

Herbst, W., Bailer-Jones, C. A. L., & Mundt, R. 2001, ApJ, 554, L197

Herbst, W., Bailer-Jones, C. A. L., Mundt, R., Meisenheimer, K., & Wackermann, R. 2002, A&A, 396, 513

Hillenbrand, L. A. 1997, AJ, 113, 1733

Hosokawa, T., Oﬀner, S. S. R., & Krumholz, M. R. 2011, ApJ, 738, 140

Howe, R. 2009, Living Reviews in Solar Physics, 6

Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398

H¨unsch, M., Weidner, C., & Schmitt, J. H. M. M. 2003, A&A, 402, 571

H¨unsch, M., Randich, S., Hempel, M., Weidner, C., & Schmitt, J. H. M. M. 2004, A&A, 418, 539

Iben, I., Jr. 1965, ApJ, 141, 993

Iglesias, C. A., & Rogers, F. J. 1996, ApJ, 464, 943

Irwin, J., Berta, Z. K., Burke, C. J., et al. 2011, ApJ, 727, 56

Irwin, J., Aigrain, S., Bouvier, J., et al. 2009, MNRAS, 392, 1456

Irwin, J., & Bouvier, J. 2009, IAU Symposium, 258, 363

Irwin, J., Hodgkin, S., Aigrain, S., et al. 2007, MNRAS, 377, 741

266 Israelian, G., Delgado Mena, E., Santos, N. C., et al. 2009, Nature, 462, 189

Jackson, R. J., & Jeﬀries, R. D. 2013, MNRAS, 431, 1883

Jackson, R. J., & Jeﬀries, R. D. 2014, MNRAS, 441, 2111

Jackson, R. J., & Jeﬀries, R. D. 2014, MNRAS, 445, 4306

Jackson, R. J., Jeﬀries, R. D., & Maxted, P. F. L. 2009, MNRAS, 399, L89

Jeﬀries, R. D. 1999, MNRAS, 309, 189

Jeﬀries, R. D. 2000, Stellar Clusters and Associations: Convection, Rotation, and Dynamos, 198, 245

Jeﬀries, R. D., Jackson, R. J., James, D. J., & Cargile, P. A. 2009, MNRAS, 400, 317

Jeﬀries, R. D., & James, D. J. 1999, ApJ, 511, 218

Jeﬀries, R. D., Totten, E. J., Harmer, S., & Deliyannis, C. P. 2002, MNRAS, 336, 1109

Johnson, H. L., & Knuckles, C. F. 1955, ApJ, 122, 209

Jones, B. F. 1972, ApJ, 171, L57

Jones, B. F., Fischer, D., & Soderblom, D. R. 1999, AJ, 117, 330

Kamai, B. L., Vrba, F. J., Stauﬀer, J. R., & Stassun, K. G. 2014, AJ, 148, 30

Kawaler, S. D. 1988, ApJ, 333, 236

Koenigl, A. 1991, ApJ, 370, L39

King, J. R., Krishnamurthi, A., & Pinsonneault, M. H. 2000, AJ, 119, 859

King, J. R., Schuler, S. C., Hobbs, L. M., & Pinsonneault, M. H. 2010, ApJ, 710, 1610

Kippenhahn, R., & Weigert, A. 1990, Stellar Structure and Evolution, XVI, 468 pp. 192 ﬁgs.. Springer-Verlag Berlin Heidelberg New York. Also Astronomy and Astrophysics Library,

Kraft, R. P. 1967, ApJ, 150, 551

Kraus, A. L., Tucker, R. A., Thompson, M. I., Craine, E. R., & Hillenbrand, L. A. 2011, ApJ, 728, 48

Krishnamurthi, A., Pinsonneault, M. H., Barnes, S., & Soﬁa, S. 1997, ApJ, 480, 303

267 Kumar, P., & Quataert, E. J. 1997, ApJ, 475, L143

Kurucz, R. L. 1979, ApJS, 40, 1

Lachaume, R., Dominik, C., Lanz, T., & Habing, H. J. 1999, A&A, 348, 897

Lamia, L., Spitaleri, C., La Cognata, M., Palmerini, S., & Pizzone, R. G. 2012, A&A, 541, A158

Lanzafame, A. C., & Spada, F. 2015, A&A, 584, A30

Li, T. D., Bi, S. L., Yang, W. M., et al. 2014, ApJ, 781, 62

Lanzafame, A. C., & Spada, F. 2015, A&A, 584, A30

Libbrecht, K. G. 1988, ApJ, 334, 510

Littlefair, S. P., Naylor, T., Mayne, N. J., Saunders, E. S., & Jeﬀries, R. D. 2010, MNRAS, 403, 545

Littlefair, S. P., Naylor, T., Mayne, N. J., Saunders, E., & Jeﬀries, R. D. 2011, MNRAS, 413, L56

Lodders, K. 2003, ApJ, 591, 1220

Longland, R., Lor´en-Aguilar, P., Jos´e, J., Garc´ıa-Berro, E., & Althaus, L. G. 2012, A&A, 542, A117

L´opez-Morales, M., & Ribas, I. 2005, ApJ, 631, 1120

L´opez-Morales, M. 2007, ApJ, 660, 732

Lyng˚a, G. 1985, IAUS, 106, 143

MacDonald, J., & Mullan, D. J. 2009, ApJ, 700, 387

Macdonald, J., & Mullan, D. J. 2010, ApJ, 723, 1599

MacDonald, J., & Mullan, D. J. 2012, MNRAS, 421, 3084

MacDonald, J., & Mullan, D. J. 2013, ApJ, 765, 126

MacGregor, K. B., & Brenner, M. 1991, ApJ, 376, 204

Maeder, A., & Zahn, J.-P. 1998, A&A, 334, 1000

Malo, L., Doyon, R., Feiden, G. A., et al. 2014, ApJ, 792, 37

Margheim, S. J., Deliyannis, C. P., King, J. R., & Steinhauer, A. 2002, Bulletin of the American Astronomical Society, 34, #124.03

268 Margheim, S. J. 2007, Ph.D. Thesis,

McQuillan, A., Mazeh, T., & Aigrain, S. 2014, ApJS, 211, 24

Meibom, S., Barnes, S. A., Platais, I., et al. 2015, Nature, 517, 589

Meibom, S., Mathieu, R. D., Stassun, K. G., Liebesny, P., & Saar, S. H. 2011, ApJ, 733, 115

Mendoza, C., Seaton, M. J., Buerger, P., et al. 2007, MNRAS, 378, 1031

Mohanty, S., & Basri, G. 2003, ApJ, 583, 451

Montalb´an, J., & Schatzman, E. 2000, A&A, 354, 943

Montgomery, K. A., Marschall, L. A., & Janes, K. A. 1993, AJ, 106, 181

Morales, J. C., Ribas, I., & Jordi, C. 2008, A&A, 478, 507

Morales, J. C., Gallardo, J., Ribas, I., et al. 2010, ApJ, 718, 502

Moraux, E., Artemenko, S., Bouvier, J., et al. 2013, A&A, 560, A13

Mullan, D. J., & MacDonald, J. 2001, ApJ, 559, 353

Noyes, R. W., Weiss, N. O., & Vaughan, A. H. 1984, ApJ, 287, 769

O’dell, M. A., Panagi, P., Hendry, M. A., & Collier Cameron, A. 1995, A&A, 294, 715

O’Neal, D., Neﬀ, J. E., & Saar, S. H. 1998, ApJ, 507, 919

O’Neal, D., Neﬀ, J. E., Saar, S. H., & Cuntz, M. 2004, AJ, 128, 1802

O’Neal, D., Neﬀ, J. E., Saar, S. H., & Mines, J. K. 2001, AJ, 122, 1954

Onehag,¨ A., Korn, A., Gustafsson, B., Stempels, E., & Vandenberg, D. A. 2011, A&A, 528, A85

Pace, G., Pasquini, L., & Fran¸cois, P. 2008, A&A, 489, 403

Pace, G., Castro, M., Mel´endez, J., Th´eado, S., & do Nascimento, J.-D., Jr. 2012, A&A, 541, A150

Palla, F., & Stahler, S. W. 1992, ApJ, 392, 667

Pasquini, L., Biazzo, K., Bonifacio, P., Randich, S., & Bedin, L. R. 2008, A&A, 489, 677

269 Pecaut, M. J., & Mamajek, E. E. 2013, ApJS, 208, 9

Perryman, M. A. C., Brown, A. G. A., Lebreton, Y., et al. 1998, A&A, 331, 81

Petrov, P. P., Shcherbakov, V. A., Berdyugina, S. V., et al. 1994, A&AS, 107, 9

Pinsonneault, M. 1997, ARA&A, 35, 557

Pinsonneault, M. H., Deliyannis, C. P., & Demarque, P. 1991, ApJ, 367, 239

Pinsonneault, M. H., Deliyannis, C. P., & Demarque, P. 1992, ApJS, 78, 179

Pinsonneault, M. H., Kawaler, S. D., & Demarque, P. 1990, ApJS, 74, 501

Pinsonneault, M. H., Kawaler, S. D., Soﬁa, S., & Demarque, P. 1989, ApJ, 338, 424

Pizzolato, N., Maggio, A., Micela, G., Sciortino, S., & Ventura, P. 2003, A&A, 397, 147

Popper, D. M. 1997, AJ, 114, 1195

Prantzos, N. 2012, A&A, 542, A67

Preibisch, T., Kim, Y.-C., Favata, F., et al. 2005, ApJS, 160, 401

Press, W. H. 1981, ApJ, 245, 286

Prisinzano, L., & Randich, S. 2007, A&A, 475, 539

Randich, S., Pasquini, L., & Pallavicini, R. 2000, A&A, 356, L25

Randich, S., Pallavicini, R., Meola, G., Stauﬀer, J. R., & Balachandran, S. C. 2001, A&A, 372, 862

Randich, S., Pace, G., Pastori, L., & Bragaglia, A. 2009, A&A, 496, 441

Rebull, L. M., Stauﬀer, J. R., Megeath, S. T., Hora, J. L., & Hartmann, L. 2006, ApJ, 646, 297

Reeves, H. 1970, Nature, 226, 727

Reiners, A. 2012, Living Reviews in Solar Physics, 9, 1

Reiners, A., Basri, G., & Browning, M. 2009, ApJ, 692, 538

Rempel, M., & Schlichenmaier, R. 2011, Living Reviews in Solar Physics, 8, 3

Ribas, I. 2003, A&A, 398, 239

Ribas, I. 2006, Ap&SS, 304, 89

270 Richer, J., & Michaud, G. 1993, ApJ, 416, 312

Rogers, F. J., & Nayfonov, A. 2002, ApJ, 576, 1064

Rogers, F. J., Swenson, F. J., & Iglesias, C. A. 1996, ApJ, 456, 902

Ryan, S. G. 2000, MNRAS, 316, L35

Saar, S. H. 1996, Stellar Surface Structure, 176, 237

Saar, S. H. 2001, 11th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun, 223, 292

Saar, S. H., & Linsky, J. L. 1986, Advances in Space Research, 6, 235

Sacco, G. G., Randich, S., Franciosini, E., Pallavicini, R., & Palla, F. 2007, A&A, 462, L23

Sackmann, I.-J., & Boothroyd, A. I. 1992, ApJ, 392, L71

Salaris, M., Weiss, A., & Percival, S. M. 2004, A&A, 414, 163

Salpeter, E. E. 1954, Australian Journal of Physics, 7, 373

Saumon, D., Chabrier, G., & van Horn, H. M. 1995, ApJS, 99, 713

Schmidt, S. J., Hawley, S. L., West, A. A., et al. 2015, AJ, 149, 158

Schou, J., Antia, H. M., Basu, S., et al. 1998, ApJ, 505, 390

Schuler, S. C., King, J. R., Fischer, D. A., Soderblom, D. R., & Jones, B. F. 2003, AJ, 125, 2085

Serenelli, A. 2016, European Physical Journal A, 52, #78

Sergison, D. J., Mayne, N. J., Naylor, T., Jeﬀries, R. D., & Bell, C. P. M. 2013, MNRAS, 434, 966

Sestito, P., Randich, S., Mermilliod, J.-C., & Pallavicini, R. 2003, A&A, 407, 289

Sestito, P., Randich, S., & Pallavicini, R. 2004, A&A, 426, 809

Sestito, P., & Randich, S. 2005, A&A, 442, 615

Shen, Z.-X., Jones, B., Lin, D. N. C., Liu, X.-W., & Li, S.-L. 2005, ApJ, 635, 608

Shu, F., Najita, J., Ostriker, E., et al. 1994, ApJ, 429, 781

Skumanich, A. 1972, ApJ, 171, 565

271 Soderblom, D. R. 2010, ARA&A, 48, 581

Soderblom, D. R., Jones, B. F., Balachandran, S., et al. 1993, AJ, 106, 1059

Soderblom, D. R., Fedele, S. B., Jones, B. F., Stauﬀer, J. R., & Prosser, C. F. 1993, AJ, 106, 1080

Soderblom, D. R., King, J. R., Siess, L., Jones, B. F., & Fischer, D. 1999, AJ, 118, 1301

Soderblom, D. R., Laskar, T., Valenti, J. A., Stauﬀer, J. R., & Rebull, L. M. 2009, AJ, 138, 1292

Soderblom, D. R., Hillenbrand, L. A., Jeﬀries, R. D., Mamajek, E. E., & Naylor, T. 2014, Protostars and Planets VI, 219

Somers, G., & Pinsonneault, M. H. 2014, ApJ, 790, 72

Somers, G., & Pinsonneault, M. H. 2015, MNRAS, 449, 4131

Somers, G., & Pinsonneault, M. H. 2015, ApJ, 807, 174

Somers, G., & Pinsonneault, M. H. 2016, arXiv:1606.00004

Spite, F., & Spite, M. 1982, A&A, 115, 357

Spruit, H. C. 1982, A&A, 108, 348

Spruit, H. C. 1982, A&A, 108, 356

Spruit, H. C., & Weiss, A. 1986, A&A, 166, 167

Stahler, S. W. 1988, ApJ, 332, 804

Starrﬁeld, S., Truran, J. W., Sparks, W. M., & Arnould, M. 1978, ApJ, 222, 600

Stassun, K. G., Feiden, G. A., & Torres, G. 2014, NewAR, 60, 1

Stassun, K. G., Kratter, K. M., Scholz, A., & Dupuy, T. J. 2012, ApJ, 756, 47

Stassun, K. G., Mathieu, R. D., Mazeh, T., & Vrba, F. J. 1999, AJ, 117, 2941

Stassun, K. G., Mathieu, R. D., & Valenti, J. A. 2007, ApJ, 664, 1154

Stassun, K. G., van den Berg, M., Feigelson, E., & Flaccomio, E. 2006, ApJ, 649, 914

Stauﬀer, J. R., Barrado y Navascu´es, D., Bouvier, J., et al. 1999, ApJ, 527, 219

272 Stauﬀer, J. R., Hartmann, L. W., Fazio, G. G., et al. 2007, ApJS, 172, 663

Stauﬀer, J. R., Hartmann, L., Soderblom, D. R., & Burnham, N. 1984, ApJ, 280, 202

Stauﬀer, J. R., Schultz, G., & Kirkpatrick, J. D. 1998, ApJ, 499, L199

Stauﬀer, J. R., Jones, B. F., Backman, D., et al. 2003, AJ, 126, 833

Stauﬀer, J. R., Schultz, G., & Kirkpatrick, J. D. 1998, ApJ, 499, L199

Steinhauer, A. 2003, Ph.D. Thesis,

Strobel, A. 1991, Astronomische Nachrichten, 312, 177

Stuik, R., Bruls, J. H. M. J., & Rutten, R. J. 1997, A&A, 322, 911

Sweet, P. A. 1950, MNRAS, 110, 548

Swenson, F. J., & Faulkner, J. 1992, ApJ, 395, 654

Takeda, Y., Honda, S., Ohnishi, T., et al. 2013, PASJ, 65, 53

Talon, S. 2008, in EAS Publications Series, Vol. 32, EAS Publications Series, ed. C. Charbonnel & J.-P. Zahn, 81–130

Talon, S., & Charbonnel, C. 2003, A&A, 405, 1025

Talon, S., & Charbonnel, C. 2005, A&A, 440, 981

Taylor, B. J. 2007, AJ, 133, 370

Terndrup, D. M., Pinsonneault, M., Jeﬀries, R. D., et al. 2002, ApJ, 576, 950

Terrien, R. C., Fleming, S. W., Mahadevan, S., et al. 2012, ApJ, 760, L9

Thorburn, J. A., Hobbs, L. M., Deliyannis, C. P., & Pinsonneault, M. H. 1993, ApJ, 415, 150

Thoul, A. A., Bahcall, J. N., & Loeb, A. 1994, ApJ, 421, 828

Tognelli, E., Degl’Innocenti, S., & Prada Moroni, P. G. 2012, A&A, 548, A41

Torres, G., & Ribas, I. 2002, ApJ, 567, 1140

Torres, C. A. O., Quast, G. R., da Silva, L., et al. 2006, A&A, 460, 695

Torres, G., Andersen, J., & Gim´enez, A. 2010, A&A Rev., 18, 67 van Leeuwen, F., Alphenaar, P., & Meys, J. J. M. 1987, A&AS, 67, 483

273 van Saders, J. L., & Pinsonneault, M. H. 2013, ApJ, 776, 67

Viana Almeida, P., Santos, N. C., Melo, C., et al. 2009, A&A, 501, 965

Villanova, S., Carraro, G., & Saviane, I. 2009, A&A, 504, 845

Ventura, P., Zeppieri, A., Mazzitelli, I., & D’Antona, F. 1998, A&A, 331, 1011

Vogel, S. N., & Kuhi, L. V. 1981, ApJ, 245, 960

Weber, E. J., & Davis, Jr., L. 1967, ApJ, 148, 217

Weymann, R., & Sears, R. L. 1965, ApJ, 142, 174

Wilson, O. C. 1966, ApJ, 144, 695

Yee, J. C., & Jensen, E. L. N. 2010, ApJ, 711, 303

Zahn, J.-P. 1974, Stellar Instability and Evolution, 59, 185

Zahn, J.-P. 1992, A&A, 265, 115

Zahn, J.-P., Talon, S., & Matias, J. 1997, A&A, 322, 320

Zappala, R. R. 1972, ApJ, 172, 57

274