Precise spectroscopic analysis of solar-type Stars with moderate and fast rotation

Precise spectroscopic analysis of solartype stars with moderate and fast rotation Maria Tsantaki Tese de Doutoramento apresentada à Faculdade de Ciências da Universidade do Porto, Departamento de Física e Astronomia 2015 Precise spectroscopic analysis of solar-type stars with moderate and fast rotation Maria Tsantaki PhD FCUP ANO 3.º CICLO P D hD Ph D Precise spectroscopic analysis of solar-type stars with moderate and fast rotation Maria Tsantaki Centro de Astrof´ısica da Universidade do Porto Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias, Universidade do Porto Tese de Doutouramento Orientadores: S. G. Sousa, N. C. Santos October, 2014 To Astrophysics “May it help free minds in destroying ignorance, superstitions, and beliefs in god-like entities.” Abstract In this thesis, I describe the processes of deriving the fundamental stellar parameters for solar-type stars. The purpose of this thesis is to optimize existing methods but also create new methodologies for determining stellar parameters that covers a very diverse group of stars. In Chapters 2 and 3, I describe the standard method of deriving stellar parameters for slowly rotating FGK stars that is based on the measurement of equivalent width (EW) of iron lines and by imposing ionization and excitation balance. Even though, this method has been successfully applied to large sample of stars, it has been reported considerable discrepancies for the lower temperature regime (Teff <5000 K). These stars have line-crowded spectra and the precise measurement of the EW is difficult. We dealt with this problem by carefully selecting a line list, using a K-type star as a reference. The new parameters for cool stars are now in agreement with more modelindependent methods, namely the infrared flux method. The new line list is also used for giant stars and for the cooler planet hosts. A principal part of this thesis was to create a procedure to deal with stars with high rotational velocities. The parameters for these stars cannot be derived with the standard EW method because their spectral lines are broadened and therefore strongly blended. In chapter 4, I present the basic principles of the spectral synthesis technique. In chapter 5, I present a refinement of the spectral synthesis technique designed to treat fast rotating stars better. The comparison of our stellar parameters shows good agreement with literature values, both for slowly and for fast rotating stars. In addition, our results are on the same scale as the parameters derived from the EW method, presented in our previous work. We applied the new methodology to transit planet hosts, as these stars have wide dispersion in their rotational velocities compared to the radial velocity targets. With this thesis, we provide the tools to derive precise stellar parameters. This work is expected to have strong impact on the study of stellar populations of the Milky Way, the characterization of planets and their hosts, and understanding stellar atmospheres. I VIII CONTENTS LIST OF FIGURES 1.1 Planetary masses presented by the year of discovery according to exoplanet.eu . . . . 16 1.2 An artist conception of known planets that are likely to be habitable (Credit to UPR Arecibo). ..................................... 17 1.3 A model grid for solid planets from 0.1 through 100 earth masses. . . . . 18 1.4 Frequency of giant planets as a function of metallicity and mass of the HARPS and the CORALIE sample. . . . . . . . . . . . . . . . . . . . . . 20 1.5 Abundance ratios vs. iron metallicity for the total HARPS sample. . . . . 22 2.1 A simple depiction of the equivalent width. . . . . . . . . . . . . . . . . . 26 2.2 The EW dependence with Teff . ........................ 28 2.3 Typical curve of growth from a model photosphere . . . . . . . . . . . . . 30 2.4 Curve of growth for different values of ξt................... 31 2.5 A schematic of the standard procedure. . . . . . . . . . . . . . . . . . . . 33 2.6 A Gaussian fit of a spectral line as given by ARES. . . . . . . . . . . . . . 35 2.7 Results of ARES for a sample of HARPS spectra. . . . . . . . . . . . . . . 36 2.8 Fe iabundance vs. excitation potential and reduced EW. . . . . . . . . . . 37 2.9 HR diagram for the Adibekyan et al. (2012) sample . . . . . . . . . . . . 38 3.1 Comparison of Teff values measured with four different methods from Molenda- ˙ Zakowicz et al. (2013). . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Different spectra for a hot and a cool star . . . . . . . . . . . . . . . . . . 42 3.3 Comparison for Teff with the EW method of Sousa et al. (2008) and with theIRFM.................................... 43 3.4 Upper panel: Curve of growth for both line lists for the reference star, HD 21749. Lower panel: The Fe iabundance versus excitation potential . 46 3.5 Reduced EW versus excitation potential for the line list of Tsantaki et al. (2013) (filled circles) and for SO08 (crosses). . . . . . . . . . . . . . . . . . 47 3.6 Comparison between temperature derived with the cool line list of this work and the results of SO08 . . . . . . . . . . . . . . . . . . . . . . . . . 49 IX XLIST OF FIGURES 3.7 Comparison between surface gravity derived with the cool line list of this work and the results of SO08 . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.8 Comparison between metallicity derived with the cool line list of this work andtheresultsofSO08 ............................ 51 3.9 Effect of temperature on the other parameters: metallicity (left panel) and surface gravity (right panel). . . . . . . . . . . . . . . . . . . . . . . . 52 3.10 Comparison of the surface gravities derived from spectroscopy of this work and from Hipparcos parallaxes ........................ 55 3.11 Trigonometric log gHIP minus loggspec as a function of temperature. . . . 55 3.12 HR diagram for the Bensby et al. (2014) sample . . . . . . . . . . . . . . 56 3.13 Comparison between the temperatures derived from this work and the IRFM....................................... 57 3.14 Comparison between the difference in temperatures derived from IRFM - This Work and IRFM - SO08. . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.15 Comparison between the spectroscopic, the IRFM and the direct temperature measurements with the literature. . . . . . . . . . . . . . . . . . . . 60 3.16 Cr and Ti ratios as a function of effective temperature. . . . . . . . . . . . 63 3.17 Comparison between our baseline parameters with those in the Extrasolar PlanetsEncyclopedia.............................. 65 3.18 Results from the TS13–SO08 line list versus the Hekker & Mel´endez (2007) 66 4.1 Radiation from a surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Observed and over-plotted synthetic spectra for some wavelength intervals. 77 4.3 Impact on the temperatures and metallicities of fixing log gto the photometric values, for three different methods. . . . . . . . . . . . . . . . . . . 78 5.1 Solar absorption lines, broadened by υsin i10 km s−1, 15 km s−1and 20 km s−1. 81 5.2 Comparison between the parameters derived using the spectral synthesis and the results of our EW method: temperature (top panel), metallicity (middle panel) and surface gravity (bottom panel). . . . . . . . . . . . . . 90 5.3 Comparison of surface gravity derived from the transit fit with this work andtheEWmethod............................... 91 5.4 Differences in temperature vs. υsin i...................... 92 5.5 Differences in surface gravity vs. υsin i. ................... 93 5.6 Differences in metallicity vs. υsin i....................... 94 5.7 Differences in stellar mass (top panel) and radius (bottom panel) vs. υsin i. 95 5.8 Temperature (top panel), surface gravity (middle panel), and metallicity (bottompanel).................................. 97 5.9 Differences in temperature (top panel), surface gravity (middle panel) and metallicity (bottom panel) versus rotational velocity for moderate/high rotators. ......... 98 5.10 Examples of synthesis fitting of HD 210302 (υsin i= 13.68 km s−1) and HD 30652 (υsin i = 17.01 km s−1). .................................. 99 5.11 Same as Fig. 5.10. .................................100 LIST OF FIGURES XI 5.12 The differences refer to surface gravity derived from a transit light curve analysis minus other methods. . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.13 Comparison between the literature data of planetary mass, the radii ratio (Rp/Rstar), and planetary radius and this work. . . . . . . . . . . . . . . . 104 5.14 Comparison between stellar density derived from the transit light curve analysis and literature data. ..................................105 5.15 Blue squares represent planetary mass and radius derived in this work in comparison with literature values (green circles). . . . . . . . . . . . . . . 106 6.1 Comparison of the effective temperature derived from Sousa et al. (2010) forhotstars...................................109 A.1 Correlation of microturbulence with temperature and surface gravity. . . . 114 B.1 Spectral synthesis for solar values using atomic data from VALD and after solarcalibration.................................116 B.2 SMEmainmenu ................................117 B.3 SME interface for setting the minimization procedure. . . . . . . . . . . . 118 B.4 An example of the SME output for HD103774. . . . . . . . . . . . . . . . 119 D.1 Comparison of stellar parameters for the benchmark stars using the GES linelist .....................................126 XII LIST OF FIGURES LIST OF TABLES 3.1 Characteristics of the sample and the reference star. . . . . . . . . . . . . 44 3.2 Mean errors in the parameters when dividing them in temperature ranges. 45 3.3 Sample of the line list used for the spectroscopic analysis. . . . . . . . . . 48 3.4 Results of the internal comparison for the whole sample and for stars cooler and hotter than 5000 K. . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5 Comparison between the effective temperatures derived with different methods. .................................... 58 3.6 Interferometric data and derived temperatures for stars in common with our sample. Stars with alternative angular diameters are also presented here........................................ 60 3.7 Updated stellar parameters for previously analyzed planet hosts. . . . . . 62 5.1 Spectral wavelength intervals and line data used for the spectroscopic analysis...................................... 83 5.2 Internal error analysis for each spectral type and different rotational velocities. ..................................... 86 5.3 Errors summed Quadratically for each spectral type and for the different rotationalvelocities. .............................. 87 5.4 Differences in stellar parameters between this work and the EW method forthe48samplestars. ............................ 88 5.5 Differences in parameters derived with different methods. . . . . . . . . . 96 5.6 Observation log of the transit hosts analyzed in this work. . . . . . . . . . 101 5.7 Spectroscopic parameters of planet hosts derived in this work. . . . . . . . 101 5.8 Transit fit parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 C.1 Results of the comparison between this work and the EW method for dwarf stars. The stars in boldface are analyzed in Sect. 5.3. . . . . . . . . 122 C.2 Results of the comparison between this work and the EW method for giant stars. The stars in boldface are analyzed in Sect. 5.3. . . . . . . . . 123 XIII XIV LIST OF TABLES C.3 Stellar parameters for a sample of fast rotating FGK dwarfs. . . . . . . . 124 E.1 The complete line list used for the spectroscopic analysis of Chapter 3. . . 127 CHAPTER 1 Introduction “The universe is a pretty big place. If it’s just us, seems like an awful waste of space.” Carl Sagan, Contact Is there anybody out there? There is no better place to answer this question but the field of extrasolar planets. Astronomers focus on exploring what would be one of the greatest discoveries of modern astronomy, new earths that could harbor life. As of January 2015, there are more than 1187 exoplanetary systems discovered1(Fig. 1.1) and the future looks even more promising as new missions are dedicated to this purpose (e.g., CHEOPS, TESS, PLATO 2.0). From the very first discovery of an extrasolar planet orbiting a solar-type star (Mayor & Queloz 1995), it became clear that our Solar system is not the only possible configuration. This new planet was found in close orbit to the host star in contrast to our system, suggesting alternative formation mechanisms than it was previously thought. The discoveries that subsequently followed, revealed an impressive diversity of planets. For example, Kepler-16 b is a Saturn-mass planet orbiting a binary star, closely resembling Tatooine from the Star Wars series, proving that nature can reproduce fiction in 1according to www.exoplanet.eu 15 16 CHAPTER 1. INTRODUCTION Figure 1.1: Planetary masses presented by the year of discovery according to exoplanet.eu the best way. An artist conception of known planets that are likely to be habitable is shown in Fig. 1.2. Most planet detection methods are indirect and focus on the observations of the host star. The most efficient ones are the radial velocity (RV) and the transit techniques. The RV method is based on the detection of variations in the radial velocity of the star, due to the gravitational pull from a planet as it orbits the star. When the star moves towards our line of sight, its spectrum is blueshifted, while it is redshifted when it moves away. While the above methods provide information about the planetary mass, the transit method can determine the radius of a planet. If a planet transits in front of the disc of its parent star, then the observed flux of the star drops a small amount. The derivation of the planetary mass and radius (and other planetary parameters that depend on the above, such as density and temperature) are critically dependent on the fundamental atmospheric parameters of the planet host. Moreover, the fast growing samples of planet hosts have revealed interesting correlations between various parameters of planets and their hosts (some of them listed in Sect. 1.2) that provide the necessary constrains on their formation and evolution theories. Even though our motivation is to study planet hosts in order to understand the planets themselves, there are even many more fields in astronomy related to the study of solar-type stars in our Galaxy. In the following sections, we present briefly the fields where fundamental stellar parameters are important. 1.1. PLANETARY AND STELLAR CHARACTERIZATION 17 Figure 1.2: An artist conception of known planets that are likely to be habitable (Credit to UPR Arecibo). 1.1 Planetary and stellar characterization To understand the physical processes involved in the formation and evolution of planetary systems, precise measurements of the fundamental properties of the exoplanets and their hosts are required. From the analysis of the light curve of a transiting planet, the planetary radius is always dependent on the stellar radius (Rp∝R?). Moreover, the mass of the planet, or the minimum mass in case the inclination of the orbit is not known, is calculated from the RV curve only if the mass of the star is known (Mp∝M2/3 ?). On the other hand, the stellar mass and radius depend on the observationally determined atmospheric parameters (with the exception of stars with interferometric measurements or stars that belong to eclipsing binaries) such as, effective temperature (Teff), surface gravity (log g), and metallicity ([Fe/H], where iron is usually used as a proxy). The latter parameters are used to deduce stellar mass and radius either from calibrations (Torres et al. 2010; Santos et al. 2013) or stellar evolutionary models (e.g., Girardi et al. 2002). It is therefore, imperative to derive precise and accurate stellar parameters to avoid the propagation of errors in the planetary properties. For instance, Torres et al. (2012) compared stellar parameters derived from spectral synthesis techniques with the ones after constraining surface gravity derived from the transit light curve. The authors show considerable systematic errors in the planetary mass and radius between the constrained and unconstrained analyses. In particular, the overestimated values of stellar radius (corresponds in turn to overestimated planetary radius) observed with the unconstrained analysis, may explain part of the anomalously inflated radii that has been reported for some Jovian planets, such as in the cases of HD 209458 b (Burrows et al. 2000) and 24 CHAPTER 1. INTRODUCTION •In addition, the infrared flux method (IRFM) proposed by Blackwell & Shallis (1977) provides precise temperature estimations based on the fact that the bolometric flux depends on the angular diameter and the effective temperature, as described by the Stefan-Boltzmann law, whereas the monochromatic flux in the infrared (IR) depends on the angular diameter but weakly on the effective temperature, this way the dependence on the angular diameter disappears: fbol fλIR =σT4 eff fλIR(model),(1.3) where fbol is the measured bolometric flux, fλIR is the measured monochromatic IR flux and fλIR(model) is the monochromatic flux in the IR derived by the assuming model. This technique is more model-independent compared to the aforementioned spectroscopic techniques and has been implemented by various authors over the years (e.g., Ram´ırez & Mel´endez 2005; Casagrande et al. 2006, 2010). The above methods are indirect which means that they require model dependencies. In lack of direct methods applicable to most stars, we have to rely to them for the parameter determinations. The different analysis techniques often yield significant differences in their results (e.g., Torres et al. 2008; Bruntt et al. 2012; Molenda- ˙ Zakowicz et al. 2013). These systematic errors are difficult to assess and are usually the main error contributors within a study. Such problems can be mitigated by a uniform analysis that will yield the precision needed. 1.5 This work In this thesis, we will focus on the derivation of photospheric stellar parameters for solar-type stars. The technique can be applied to both high and medium resolution spectroscopy. The first Chapters describe the method based on the ionization and excitation balance of iron. The EW method built by our team was problematic for stars with low Teff (below 5000 K). Our goal was to optimize this method for the cooler stars and re-derive their correct stellar parameters. In the second part of the thesis we use another spectroscopic technique, namely spectral synthesis. We describe the principles of our methodology (e.g., code, procedure). Our aim is to provide stellar parameters for stars where the EW technique is not effective. We have applied this new method to stars with moderate and high rotation. This could be the case of transit planet hosts since the transit technique is not limited by stellar rotation, at least in first approach. Planet hosts with moderate and high rotation were analyzed with our method and their planetary parameters were revised. CHAPTER 2 Spectroscopic stellar parameters - EW method “It’s a dirty job but someone has to do it.” Anonymous In this section, we describe the procedure to derive stellar parameters for solar-type stars by measuring the equivalent widths (EW) of iron lines and by forcing the ionization and excitation balance. The theory behind this procedure is depicted in detail in Gray (2005). 2.1 The equivalent width An important characteristic of FGK stars is the presence of many absorption features in their spectra because the atoms and molecules are not fully ionized. A common approach to derive atmospheric parameters for these spectral types is to exploit the large number and properties of the iron lines. In optical spectra iron lines are the most numerous with well-studied atomic transitions that make the abundance determination easier compared to other elements. Usually, we refer as metallicity to the amount of iron content in a star. This is not strictly correct as for example, in the Sun, there are more abundant elements than 25 26 CHAPTER 2. THE EW METHOD Figure 2.1: A simple depiction of the equivalent width. Taken from Pradhan & Nahar (2011). iron (C, N, O, Ne, Mg, and Si). However, the overall metallicity correlates with iron abundance for field stars (Bodaghee et al. 2003; Gilli et al. 2006) and if we exclude the chemically peculiar stars, we can safely assume iron as a proxy for the overall metallicity. The EW of a spectral line provides a measure of its strength. The EW forms a rectangle with a height equal to that of the continuum and a width such that the area of the rectangle is equal to that absorbed by the spectral line (Fig. 2.1). The mathematical description is given below: EW = +∞ Z −∞ Fc−Fλ Fc dλ, (2.1) where Fcis the flux of the continuum and Fλis the line flux at each wavelength λ. The strength of the line depends on the atomic transition, the absorption coefficient, and the number of absorbers which in turn, depend on temperature, electron pressure and atomic constants (see more in the following Sections). 2.2 The temperature dependence The spectral line strength is strongly correlated with temperature mostly due to the dependence on the ionization and excitation processes described below. We assume that the standard thermodynamic relations hold locally, despite the temperature and pressure gradients in the atmosphere, as described by the local thermodynamic equilibrium (LTE) approximation. In this regime, we consider that collisions (rather that radiation) dominate the excitation of the atoms, which is a good approximation in the case of FGK 2.2. THE TEMPERATURE DEPENDENCE 27 atmospheres. We can express the ratio between the number of atoms in an energy level nand the total number of the atoms of that species as: Nn N=gn u(T)10−Θ(T)χn,(2.2) where Nnis the population of energy level n,Nis the total number of atoms, gnis the degeneracy of level n,χnis the excitation potential of the same level, T is the temperature, Θ(T) = 5040/T,u(T) = Pgie−χi/kT is the partition function, and k is the Boltzmann constant. Similarly, the ionization for the collision-dominated gas can be calculated using the Saha Equation: Ni+1 Ni =Φ(T) Pe Φ(T) = (πme)3/2(5kT)5/2 ~3 ui+1(T) ui(T)e−I/kT (2.3) where the Ni+1/Niis the ratio of the total populations of atoms in two ionization states, iand i+1, the ui+1(T)/ui(T) is the ratio of partition functions, meis the electron mass, ~is the reduced Planck constant, Peis the electron pressure, and Iis the ionization potential. The typical behavior of a weak, metal line with effective temperature is shown in Fig. 2.2, for neutral and ionized species. In the atmospheres of FGK dwarfs, most metals are singly ionized and therefore, the dependence of the EW with Teff is presented by Case 2 (neutral species, where element is mostly ionized) and Case 4 (ionic species, where element is mostly ionized). In the same Figure, Case 1 depicts the behavior of the EW of neutral species with the element mostly neutral and Case 4, of ionized species where the element is mostly ionized. The strength of a weak line (R) is proportional to the ratio of the line to continuous absorption coefficients and is defined as: R=Fc−Fλ Fc = constant lv κv ,(2.4) where Fcis the flux of the continuum, Fλis the line flux at each wavelength λ,lvis the line absorption coefficient, and κvthe continuous absorption coefficient (see Chapter 4 for a definition of the latter parameters). The strength of a weak line has a direct connection with the EW as seen from Equation 2.1. The fractional change of Rwith temperature (T) shows its sensitivity to T. The behavior of a neutral species is described as: 1 R dR dT =χ+ 0.75 −I kT 2,(2.5) where the sensitivity to temperature depends on the lower-level excitation potential of 28 CHAPTER 2. THE EW METHOD Figure 2.2: The EW dependence with effective temperature. S0= 1.0 refers to the solar value of Teff (=5777 K). Taken from Gray (2005). the line (χ). For solar-type stars, usually χ+ 0.75 < I for metals, therefore, the EW of neutral species decreases with with increasing temperature. An effective way to constrain the stellar temperature is to use the excitation equilibrium of neutral lines of different χ. For ionic species, the dependence has an extra dependence on the electron pressure: 1 R dR dT =2.5 T+χ+ 0.75 kT 2−Ω.(2.6) where Ω can be considered as a constant as it mildly depends on pressure. Ionic species show smaller sensitivity to temperature changes, except for those with high excitation potential. The temperature dependence for strong lines should include the damping constants. 2.3 The pressure dependence Pressure dependence in stellar atmosphere can be related to gravity dependence. For FGK stars, an increase in the surface gravity compresses the photosphere, resulting in an 2.4. THE ABUNDANCE DEPENDENCE 29 increase in pressure (both electron and gas). The electron pressure is much smaller than the gas pressure because hydrogen, as a main electron contributor, is not fully ionized. An empirical approximation of gas pressure (Pg) in dependence with gravity (g) is given for cool stars: Pg≈constant g2/3(2.7) Also, electron pressure (Pe) is described by: Pe≈constant g1/3(2.8) In solar-type stars, where the elements are mostly ionized, NnNn+1 and the total number of atoms, N, equals to Nn+1. From the Saha equation (Equation 2.3), we obtain Nn= constant Pe. The line strength for neutral atoms is: R=lv κv≈constant Nn constant Pe≈constant Pe Pe≈constant.(2.9) Therefore, neutral atoms are insensitive to pressure changes. For first ions, we have the opposite population, Nn+1 =N. The line strength for ions becomes: R=lv κv≈constant N constant Pe≈constant g1/3.(2.10) Obviously ions are pressure sensitive, with lower pressure causing stronger lines. In addition, the wings of strong lines are good pressure indicators. This sensitivity arises from the pressure dependence of the damping parameters, namely the van der Waals and Quadratic Stark constants (see Chapter 4). 2.4 The abundance dependence As the abundance increases, line strength also increases but not always linearly. The dependence of the EW with the abundance is described by the curve of growth, as shown in Fig. 2.3, and is divided into three different regimes. The first one corresponds to the behavior of weaker lines, where the Doppler core dominates and the EW is proportional to the abundance A. The second phase begins when the central depth approaches the maximum value. The line saturates and grows asymptotically towards a constant value. The third one starts as the optical depth of the line becomes significant compared to the absorption of the continuum and the wings dominate the line profile. In this case, the EW is proportional to the square root of the abundance. It is clear that for abundance determinations, we want to select weak lines that fall on the linear part of the curve of growth, where the EW is more sensitive to abundance changes. 30 CHAPTER 2. THE EW METHOD Figure 2.3: a) Typical curve of growth from a model photosphere: the reduced EW versus abundance (A). b) Line profile change with chemical abundance of the absorption species. The dots in (a) correspond to the different profiles in (b). Taken from Gray (2005). 2.5 Microturbulence Microturbulence (ξt) is a parameter that describes the small-scale mass motions in dimensions of the optical depth. The velocities due to these motions produce Doppler shifts analogous to the thermal motions and are postulated by Gaussian distributions. Therefore, absorbers have additional turbulent velocities than the thermal ones, causing broadening with a wavelength shift of ∆λ, ∆λ=λ c2kT m+ξ21/2 ,(2.11) where λis the central wavelength, Ttemperature, kBoltzmann constant. This shows that the line affected by ξtis broadened as if the temperature is increased. In the abundance analysis, microturbulence is introduced to reconcile differences between the observed EW of saturated lines and the ones predicted from the classic 2.6. MODEL ATMOSPHERES 31 Figure 2.4: Curve of growth for different values of microturbulence. Taken from Gray (2005). one-dimensional models (static and radial). The mechanism that is believed to be responsible for this observed velocity is convection, both in low-mass stars and massive stars. Figure 2.4 shows that the presence of ξtcauses a delay in saturation. The observational data show a dependence of ξton both effective temperature and on surface gravity (e.g., Nissen 1981; Reddy et al. 2003; Allende Prieto et al. 2004; Adibekyan et al. 2012a; Ram´ırez et al. 2013). 2.6 Model atmospheres The spectroscopic techniques for parameter determinations are model dependent. This means that a model photosphere has to be constructed under some assumptions in which temperature and pressure are calculated as a function of the optical depth. In particular, a model atmosphere describes the depth-dependence of basic physical quantities: opacity at some reference frequency, electron temperature, electron pressure, gas pressure, abundances of different elements. The computation of a model atmosphere is simplified under the certain assumptions (see more in Sect. 4.1): •Homogeneous plane-parallel layers The geometrical thickness of a stellar atmosphere is sufficiently small compared to the stellar radius. In case of the Sun, geometrical thickness of the photosphere is less than 0.1% in ratio. The plane-parallel assumption is appropriate for most main-sequence and giant stars, but will fail for the super giants, where they have large atmospheric extensions (>5%), or for stars with fast expanding envelopes. Homogeneity in the atmosphere requires that the physical quantities vary only 32 CHAPTER 2. THE EW METHOD with depth and magnetic fields, star spots and granulation are ignored. •Hydrostatic equilibrium Hydrostatic equilibrium may also be assured for main-sequence and giant stars. The atmospheres of these stars do not show large-scale gas motions such as expansion or contraction. Gravity determines the pressure profile, assuming a star at equilibrium or not in a quickly evolving evolutionary stage. •Time independent The atmosphere is stationary and the properties do not change with time. The phenomena such as rotation, pulsation, expanding envelopes, variable magnetic fields, etc. are neglected. In this case, the radiative transfer equation has no time dependence (see Sect. 4.1). •Radiative equilibrium Radiative equilibrium states that the bolometric flux in a plane-parallel atmosphere is constant. In some stars, particularly in late-type stars, convective energy transportation becomes important and we need to take into account. •Local thermodynamic equilibrium As we mentioned in Sect. 2.2, it is assumed that all thermodynamic properties in a small volume have the thermodynamic equilibrium values at the local values of temperature and pressure. A system is in LTE if the local kinetic temperature is equal to the Planckian temperature of the radiation field. Usually, a model atmosphere is presented in a tabular form where some physical properties, such as local temperature, pressure, and density are listed for each atmospheric layer. The optical depth (i.e. the layers of the atmosphere) is normally chosen at a wavelength in the visible region of the spectrum, e.g. at 5000 ˚ A. In the literature there are many atmospheric models precomputed in grids for a set of stellar parameters (e.g., ATLAS - Kurucz 1993, MARCS - Gustafsson et al. 2008). 2.7 The procedure for the EW method The standard determination of spectroscopic parameters (Teff , [Fe/H], log g, and ξt) for solar-type stars starts by measuring the EW of selected and well-defined absorption lines. Then we translate these measurements into individual line abundances, assuming a given atmospheric model. We obtain the correct stellar parameters by imposing excitation and ionization balance for the iron species. The procedure of the standard method is shown in Fig. 2.5. The main steps are identified below: •First, we define a list of neutral and ionized iron lines. For these lines precise atomic data are needed. 2.7. THE PROCEDURE FOR THE EW METHOD 33 ARES ARES Automatic EWs Interpolation Minimization algorithm MOOG MOOG Spectrum Line list FeI, FeII ~300 for GF ~150 for K Atomic data EW measurements Model Atmospheres (Kurucz) Best parameters Teff , logg, [Fe/H], and micro Figure 2.5: A schematic of the standard procedure. •The EW of the selected lines should be measured precisely. The uncertainties of the EW depend on the determination of the continuum position, the signal-tonoise (S/N) of the data, the presence of blended lines, or any unexpected spectral feature, such as cosmic rays hits. Typical uncertainties of the EW values are of 2-5% for S/N ∼100. •A stellar atmospheric model is interpolated from a grid of precomputed models for a set of initial stellar parameters. •The abundances of individual iron lines are computed. •The best parameters are obtained when the Fe iabundance shows no dependence on the excitation potential (excitation balance) and on the reduced equivalent width. Additionally, the mean abundances given by Fe iand the Fe ii must be the same (ionization balance) and consistent with those of the input model atmosphere. 40 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS solar-type stars, the effective temperatures obtained with these methods can differ significantly (e.g., Kovtyukh et al. 2003; Ram´ırez & Mel´endez 2004; Casagrande et al. 2006; Sousa et al. 2008; Molenda- ˙ Zakowicz et al. 2013). A comparison of different methods is shown in Fig. 3.1, where the authors compared four different methods of parameter determinations for a sample of 169 solar-type stars finding considerable differences in their results (Molenda- ˙ Zakowicz et al. 2013). For instance the mean differences in Teff between spectroscopic (ROTFIT - Frasca et al. 2003, ARES+MOOG - Sousa et al. 2007, 2008, VWA - Bruntt et al. 2012) and photometric methods (IRFM - Pinsonneault et al. 2012) are ∼100 K. The temperature determination becomes more difficult when we focus on K-type stars. Part of the difficulties in the stars with Teff <5000 K emerge from their line crowed spectra that cause strong blending. Blending can be a considerable problem if one uses the technique based on the iron EWs. The spectral lines cannot be easily resolved and the continuum placement becomes more difficult, causing bad measurement of the EWs and hence, makes the calculation of stellar parameters ambiguous (Fig. 3.2). Therefore, it is important to carefully select the iron lines in such manner that will eliminate the blending effects. In addition, the choice of the atomic parameters influences the abundance determination. Some authors calculate the atomic parameters using the Sun as a reference to avoid the errors that emerge from the theoretical or laboratory values. For instance, an error of 5% in the atomic parameters, namely the oscillator strength, propagates to a 2% error in Teff (Allende Prieto et al. 1998). In that way, the atomic parameters for stars that are different from the Sun, i.e. too hot or too cool are no longer accurate enough. Sousa et al. (2008) (hereafter SO08) performed a spectroscopic analysis for a sample of solar-type stars. This sample is part of the High Accuracy Radial velocity Planet Searcher (HARPS) guaranteed time observations (GTO) survey that is composed of slow rotators and low activity FGK stars in order to detect low-mass planets. A comparison of these spectroscopic results with the infrared flux method indicates a disagreement in the effective temperatures only for the cooler stars of the sample with temperatures below ∼5000 K. Figure 3.3 shows the effective temperatures derived by the spectroscopic analysis and the IRFM in the work of SO08. To recover the bolometric flux that is missing from the multi-band photometry for the IRFM, the authors used two different models: 1) the ATLAS9-ODFNEW models (Castelli & Kurucz 2004) (upper panel), and 2) the Phoenix models (Brott & Hauschildt 2005) (bottom panel). Motivated by that, we compile an optimized line list to improve the accuracy of the stellar parameters for the cooler stars and compare our results with other independent methods (IRFM, interferometry). 3.2 Stellar sample and previous spectroscopic analysis The stellar sample, presented in SO08, is composed of 451 stars as part of the HARPS high-precision GTO program at the ESO La Silla 3.6m telescope with the objective to detect low-mass exoplanets with high radial velocity accuracy (Mayor et al. 2003). It 3.2. PREVIOUS SPECTROSCOPIC ANALYSIS 41 Figure 3.1: Comparison of Teff values measured with four different methods from Molenda- ˙ Zakowicz et al. (2013): ROTFIT and ARES+MOOG (Molenda- ˙ Zakowicz et al. 2013), VWA (Bruntt et al. 2012; Thygesen et al. 2012) and IRFM (Pinsonneault et al. 2012). In the insets, the mean difference between the compared sets of data, the standard deviation of the mean, and the number of stars in common are given. 42 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS Figure 3.2: Depiction of a cool and a hot spectrum for the same wavelength interval of stars with similar metallicity and gravity. Cooler spectra are more line crowded and the continuum is not well defined. 3.2. PREVIOUS SPECTROSCOPIC ANALYSIS 43 Figure 3.3: Comparison for the effective temperature with the EW method of Sousa et al. (2008) and with the IRFM using either the Kurucz (upper panel) or Phoenix models (bottom panel). 44 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS Table 3.1: Characteristics of the sample and the reference star, described in SO08. Teff log g[Fe/H]Mass (K) (dex) (dex)M Lowest value 4556 ±98 3.68 ±0.05 −0.84 ±0.01 0.37 Highest value 6403 ±65 4.62 ±0.03 0.39 ±0.05 1.42 HD 21749 4723 ±143 4.40 ±0.33 −0.02 ±0.08 0.76 is mainly comprised of dwarf FGK stars selected from a volume-limited sample of the CORALIE survey (Udry et al. 2000). Planet hosts from the southern hemisphere were also added to this sample, forming in total a sample of 451 stars. These stars are slowlyrotating, non-evolved, and low-activity stars, with apparent magnitudes that range from 3.5 to 10.2 and have distances of less than 56 parsec. The spectral resolution is of R∼110,000 and 90% of the combined spectra have S/N higher than 200. We point to SO08 for more details. For this sample, SO08 derived stellar parameters by imposing excitation and ionization equilibrium, based on the measurements of weak iron lines. This method is very effective for FGK stars due to the numerous iron lines in their spectra. Iron abundance is used as a proxy for the overall stellar metallicity. The line list for their spectroscopic analysis, was composed of 263 Fe iand 36 Fe ii lines. The EWs of the lines were measured automatically with ARES. The atomic parameters of the iron lines, namely the oscillator strength values (log gf), were computed by an inverted solar analysis, using a solar model with Teff = 5777 K, log g= 4.44 dex, ξt= 1.0 m s−1, log(Fe) = 7.47 dex. The spectroscopic analysis was completed assuming LTE, and using the 2002 version of the abundance determination code MOOG and a grid of Kurucz Atlas 9 model atmospheres. Some characteristics of the sample are depicted in Table 3.1, as described in SO08. There are different sources of uncertainties that occur in the stellar parameter determination using this method. These errors can be attributed to the uncertainties of the measurements of the EWs, the uncertainties in the atomic parameters and the uncertainties that are intrinsic to the method of ionization and excitation equilibrium. In addition, systematic errors can arise due to the assumptions of the method, such as 1D static atmospheres, NLTE effects (Mashonkina et al. 2011; Bergemann et al. 2012). However, departures from LTE for Fe lines do not affect their abundance determinations for near solar metallicity dwarfs but should be taken into consideration for more evolved or very metal-poor stars (Lind et al. 2012; Ruchti et al. 2013) that are not part of this sample. Errors in the measurements of the EWs can be minimized by using high quality spectra. In low S/N spectra, weak lines cannot be distinguished from noise and strong 3.3. BUILDING A STABLE LINE LIST FOR THE COOLER STARS 45 Table 3.2: Mean errors in the parameters when dividing them in temperature ranges. Temperature range Teff log g ξt[Fe/H] (K) (dex) (km s−1) (dex) Teff < Teff- 300 K 51 0.13 0.23 0.05 Teff- 300K < Teff < Teff+ 300 K 18 0.05 0.03 0.01 Teff+ 300 K < Teff 30 0.07 0.05 0.02 lines can be underestimated due to the miscalculation of their wings. The high resolution and high S/N spectra used for this sample, are the best solution to deal with such errors. Since in our spectroscopic analysis the atomic data (log gf) are derived with respect to the Sun, we expect small errors for solar analogs but more significant for cooler and hotter stars. For example, if we divide the sample of SO08 in three temperature groups: [Teff < Teff- 300 K], [Teff- 300 K< Teff < Teff+ 300 K], and [Teff+ 300 K < Teff ], the respective mean errors are shown in Table 3.21. The errors in the atmospheric parameters are estimated in a similar way as in Neuforge-Verheecke & Magain (1997) and Gonzalez & Vanture (1998) by varying each parameter (temperature, surface gravity and microturbulence) by their typical error. The error in ξtis determined from the standard deviation in the slope of the least-squares fit of Fe iabundance versus reduced equivalent width. The error in Teff is determined from the error in the slope of the least-squares fit of Fe iabundance versus χderived from the standard deviation in the slope and from the error in ξt. The error in log gcomes from the contribution from the error in Fe ii abundance due to the error in Teff and the scatter in the Fe ii abundances (measured as σ/√N,σis the standard deviation and N the number of lines). The error in the Fe abundance is a combination of the errors in Fe iabundance due to Teff,ξt, and the scatter of the individual Fe iabundances, added in quadrature. The use of many iron lines can reduce this type of uncertainty, assuming that the majority of the lines are of good quality. 3.3 Building a stable line list for the cooler stars A reliable line list is comprised of lines that can be accurately measured, which usually means unblended lines. In addition, lines must be unsaturated, cover a wide range in excitation potential and have accurate atomic data. Temperature, as well as the other stellar parameters, is strongly correlated with the EW. This sensitivity emerges from the excitation and ionization processes that follow the exponential and power dependencies with temperature that are defined by the well-known Boltzmann and Saha equations. For the cooler stars line blending is severe, which makes the measurements of the EWs problematic. In particular, blending effects cause an overestimation of the EWs as two blended lines cannot be resolved. Another bias in the EW measurements may come from the fitting of strong lines. 1Teffis the solar temperature. 46 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS 6 6.5 7 7.5 8 8.5 0 1 2 3 4 5 6 A(Fe I) (dex) Excitation potential (eV) This Work - slope 0.00 (dex/eV) SO08 - slope 0.03 (dex/eV) -6 -5.5 -5 -4.5 -4 -7 -6 -5 -4 Reduced EW (dex) log gf - Θχex This Work SO08 Figure 3.4: Upper panel: Curve of growth for both line lists for the reference star computed for the temperature (Θ=5040/Teff ) of Tsantaki et al. (2013). Circles represent the reduced EW of the line list of this work and crosses the line list of SO08. Lower panel: The Fe iabundances of the reference star versus the excitation potential. The dashed line shows the positive slope that corresponds to the line list of SO08. The solid line corresponds to the line list of this work and the slope is obviously zero. 3.3. BUILDING A STABLE LINE LIST FOR THE COOLER STARS 47 -6.5 -6 -5.5 -5 -4.5 -4 0 1 2 3 4 5 6 Reduced EW (dex) Excitation potential (eV) this work-slope -0.05 (dex/eV) SO08-slope -0.12 (dex/eV) Figure 3.5: Reduced EW versus excitation potential for the line list of Tsantaki et al. (2013) (filled circles) and for SO08 (crosses). The slope of the linear fits is also depicted. Gaussian fitting is a good approximation for weak lines and it can be reliable up to 150 m˚ A based on our experience, whereas a Voigt profile should be used for stronger lines. Saturated lines that deviate significantly from the linear part of the curve of growth should also be avoided in the abundance analysis. The EW predicted by the models of strong lines that are highly saturated, is quite dependent on microturbulence. A wrong estimation of microturbulence, will then produce errors in the abundance of any highly saturated line. On the other hand, weak lines that are strongly blended can lead to a underestimation of the continuum and consequently of the EW. This effect, however, is less significant. An overestimation in the EW due to blending, as well as the underestimation of very strong lines could be the reason for the systematic raise in temperature that is observed for the cooler stars of SO08. In addition, the reduced EW could also be affected by such biases, leading to correlations with the excitation potential and therefore a degeneracy between Teff and ξt. In Fig. 3.5, we show that such correlations with the new line list is reduced considerably when compared with the previous one. Therefore, our aim is to optimize the iron line list of SO08. With this goal, we use the K-type dwarf HD 21749 with Teff = 4723 K (see Table 3.1), as reference in order to check for unblended lines in its high S/N spectrum (∼150 at 6070˚ A). After visual inspection, we only consider weak, isolated lines that give good estimation for the local continuum. We avoid strong lines (>150 m˚ A) in order to apply Gaussian profiles. For the reference 48 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS Table 3.3: Sample of the line list used for the spectroscopic analysis with the atomic parameters of Fe iand Fe ii as well as the corresponding EWs of the reference star HD 21749. λ(˚ A) χlog gf Element EW(m˚ A) 4508.28 2.86 −2.403 Fe ii 53.0 4520.22 2.81 −2.563 Fe ii 72.4 4523.40 3.65 −1.871 Fe i101.7 4537.67 3.27 −2.870 Fe i43.2 4551.65 3.94 −1.928 Fe i41.9 4556.93 3.25 −2.644 Fe i57.9 4566.52 3.30 −2.156 Fe i68.6 4574.22 3.21 −2.353 Fe i55.1 4576.34 2.84 −2.947 Fe ii 29.6 ... ... ... ... ... star we show the curve of growth (see Gray 2005) using both line lists (Fig. 3.4 upper panel). Limiting the EW cut off, we mitigate in large amount the problem of saturated lines and microturbulence. The proof of that mitigation is the fact that the derived temperatures with the new line list agree with other less model-dependent methods (see Sect. 3.6). Very weak lines (<10 m˚ A) were also excluded so that noise is not superposed to these lines. The region of the spectrum below 4500 ˚ A is neglected due to the higher blending. The final line list is compiled with 120 Fe iand 17 Fe ii lines, as shown in Table 3.3. The complete list is available in Appendix E and in an online version2. As mentioned before, the effective temperature is derived when the correlation coefficient between log(Fe i) and χis zero. In the lower panel of Fig. 3.4, we demonstrate this correlation for the reference star using the line list of this work and of the work of SO08 using the parameters derived with the line list of this work. The positive slope for the line list of SO08 is translated as an overestimation in temperature of ∼180 K for this star. In addition, the abundances with the new line list show a smaller scatter which corresponds to smaller errors in the final temperature value. 3.4 New stellar parameters for 451 FGK stars in the HARPS GTO sample To check the effectiveness of the new line list, we re-derive stellar parameters for the 451 stars of the sample. For consistency, we use the same EWs as in SO08 that were measured automatically for all stars with the ARES code. In addition, we use 2http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/555/A150 3.4. NEW PARAMETERS FOR 451 STARS FROM HARPS 49 -400 -200 0 200 4500 5000 5500 6000 6500 This Work - SO08 Teff (K) This Work 4500 5000 5500 6000 6500 Teff (K) SO08 <∆Teff>=-31 K σ=53 K Figure 3.6: Comparison between the temperature derived with the cool line list of this work and the results of SO08. ∆Teff corresponds to this work minus SO08. Triangles represent stars with planets taken from Table 3.7 (see Sect. 3.7). the same damping parameters and atomic data for the iron lines. After a preliminary determination of the fundamental parameters, we perform a ’3σclipping’ procedure for lines that contribute with abundances higher than 3σfrom the average abundance3. This procedure was also applied in SO08. Microturbulence is used as a free parameter and is also derived from this spectroscopic analysis. The correlation of microturbulence with temperature and surface gravity is presented in Appendix A. This calibration can be useful in cases where the value of ξt is set fixed. The stellar masses are calculated using the stellar evolutionary models from the Padova group4. The errors of the fundamental parameters are internal, attributed to the method. They, thus, represent relative errors and not the absolute accuracy. 56 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS Figure 3.12: HR diagram for the Bensby et al. (2014) sample when a) log gis based on Fe i–Fe ii ionization equilibrium, and b) when log gis based on Hipparcos parallaxes. In b) the sizes of the circles are scaled with the difference between Fe iand Fe ii abundances. Red circles mark those stars where the Fe iabundances are lower than the Fe ii abundances, and vice versa for the blue circles. The Yonsei-Yale (Y2) isochrones are shown from 1 to 15 Gyr in steps of 1 Gyr. K-stars have been proposed to explain these differences even though these effects should be more evident for young stars (Morel & Micela 2004; Schuler et al. 2010). Other possible explanations for these differences have to do with the iron ionization method itself. We use Fe ii lines that strongly depend on surface gravity. For solar-type stars, however, these lines are not weak, leading to poorly constrains of log g. On the other hand, Fe ilines are numerous but insensitive to log gchanges. We have to note though, that temperatures and metallicities derived using the ionization and excitation equilibrium of iron lines are shown to be mostly independent of the ionization balance (Torres et al. 2012). Hence, the temperatures and metallicities derived with our spectroscopic method are precise, even if the derived spectroscopic surface gravities differ from the trigonometric values (see also Santos et al. 2013). Recently, Bensby et al. (2014) noted the same ionization problem in their samples. The authors derived surface gravity values using the EW method and compared with the trigonometric ones for FKG stars in their sample. As indicated by their results (see Fig. 3.12), ionization balance has its limitations, and mainly on the lower main sequence for stars with log g < 4.2 dex and Teff <5600 K. After applying a linear correction to the ionization balance parameters (Fig. 3.12c), there is a better match with the isochrone lines. 3.6 Comparison with other methods To evaluate the consistency of our results, namely for Teff , we compare them with other techniques. Here, we present a comparison with two different methods that are 3.6. COMPARISON WITH OTHER METHODS 57 -400 -200 0 200 400 4500 5000 5500 6000 6500 IRFM - This Work Teff (K) This Work 4500 5000 5500 6000 6500 Teff (K) IRFM Figure 3.13: Comparison between the temperatures derived from this work and the IRFM for stars in common. considered to be less model dependent, the infrared flux method (IRFM) and interferometry, respectively. 3.6.1 The infrared flux method - IRFM The IRFM method (Blackwell & Shallis 1977) is a semi-direct method for determining stellar parameters. The principle of this method relies on the fact that the bolometric flux depends on the angular diameter and the effective temperature, as described by the Stefan-Boltzmann law, whereas the monochromatic flux in the infrared (IR) depends on the angular diameter but weakly on the effective temperature, this way the dependence on the angular diameter disappears, described by Equation 1.3. The IRFM has the advantage that the dependence on the models is limited while the spectroscopic effective temperatures have considerable model dependence. We compare our results with the work of Casagrande et al. (2010, 2011) that implement the IRFM method for a large sample of stars. The authors estimate the bolometric flux from multi-band photometric measurements in the optical (BV(RC)C) band and in the near-IR (2MASS JHKS) band. For the missing spectral regions, the flux is calculated by synthetic spectra computed from model atmospheres. The absolute calibration of Vega is based on its synthetic spectrum with an uncertainty of the zero point of ∼15 K (Casagrande et al. 2010). 58 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS Table 3.5: Comparison between the effective temperatures derived with different methods. σrepresents the standard deviation and N the number of stars for the comparison. Method ∆Teff (K)σ(K)N IRFM – this work +33 ±3 54 347 IRFM – SO08 +14 ±3 61 347 IRFM – this work (Teff <5000 K) +42 ±12 65 29 IRFM – SO08 (Teff <5000 K) −86 ±16 86 29 Interferometry – this work −4±33 98 9 Interferometry – SO08 −56 ±35 105 9 Interferometry – IRFM −98 ±83 204 6 Figure 3.13 depicts the comparison between the spectroscopic temperatures and the IRFM for the stars in common. Temperatures of 341 stars are taken from Casagrande et al. (2011) using stars with direct application of the IRFM (irfm sample) and stars with Teff derived from color calibrations (clbr sample). For the clbr sample, the effective temperatures and bolometric fluxes were computed using the color calibrations in (b –y), (BT–VT), (VT–J), (VT–H) and (VT–KS) from Casagrande et al. (2010). Moreover, temperatures for six stars were taken from Casagrande et al. (2010). The comparison between the results of this work and the IRFM shows good agreement for all temperature ranges. In particular, for the cooler temperature region, the differences in Teff between this work and the IRFM are much smaller and more homogeneously distributed than between SO08 and the IRFM. The mean differences in temperature for the comparison samples are shown in Table 3.5. It is clear that the differences in temperature for this work with the IRFM are constant throughout the temperature range, with a small offset of 33 K for the whole sample. For the cooler stars these differences are ∆Teff = 42 ±12 K that are much smaller than SO08 with ∆Teff = -86 ±16 K. Figure 3.14 shows the comparison of stars with Teff<5000 K. It is evident that the trend in ∆Teff of SO08 mostly disappears with the new temperatures. 3.6.2 Interferometry Precise measurements of stellar angular diameters are acquired through long baseline interferometry both in the optical and in the infrared (e.g., SUSI - Davis & Tango 1986; Davis et al. 2011, Mark III - Mozurkewich et al. 1991, NPOI - Nordgren et al. 1999, IOTA - Dyck et al. 1996, PTI - Colavita et al. 1999, VLTI - Glindemann et al. 2000, CHARA - McAlister et al. 2005). The standard practice to determine the angular diameter is to fit the observed visibilities as a function of baseline to a uniform disc model. The angular size is connected to the more realistic limb darkened angular size (θLD) using correction factors from model atmospheres (Claret 2000). The effective temperature is 3.6. COMPARISON WITH OTHER METHODS 59 -400 -200 0 200 4400 4600 4800 5000 IRFM - Teff (K) Teff (K) IRFM Figure 3.14: Comparison between the difference in temperatures derived from IRFM - This Work (circles) and IRFM - SO08 (triangles). The dashed and the solid line depict the linear fits of the data with slopes: +0.20±0.04 and +0.43±0.05, respectively. then derived with the standard relation: Teff =L 4πσR21/4 =4fbol σθ2 LD 1/4 ,(3.4) where fBol is usually calculated from the Spectral Energy Distribution. We compare our results with the temperatures derived from interferometry. Unfortunately, the number of stars with available angular diameters for this sample is only down to a few since these measurements are challenging for dwarfs due to their small photospheric discs that are difficult to resolve. We have 9 stars in common for the comparison with spectroscopy and 6 with the IRFM. We use only direct angular diameters and bolometric fluxes available in the literature from Table 3.6. Our results show better agreement for these stars to interferometry than when comparing with the values of SO08 and the IRFM (see Table 3.5). We have to note though, that the comparison sample is very small and the values of Teff were derived from the clbr sample that is not the best representative for the IRFM precision. The differences in temperatures for the different methods are plotted in Fig. 3.15. For 3 stars (HD 10700, HD 26965, HD 146233) we include angular diameter measurements from different authors that give different Teff and are represented with different symbols. In the same figure, we see that from the stars with multiple measurements, the ones with the smallest uncertainty (∆θ θ%) agree better with the temperatures derived with 60 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS Table 3.6: Interferometric data and derived temperatures for stars in common with our sample. Stars with alternative angular diameters are also presented here. Star θLD ∆θ/θ TInt eff TSO08 eff Tthiswork eff TIRFM eff References HD (mas) (%) (K) (K) (K) (K) 10700 2.022 ±0.011 0.54 5383 ±47 5310 ±17 5322 ±17 5459 ±80 1, a ... 1.971 ±0.050 2.54 5449 ±83 ... ... ... 2, a 11964 0.611 ±0.081 13.25 5413 ±359 5332 ±22 5285 ±21 - 3, b 19994 0.788 ±0.026 3.30 6109 ±111 6289 ±46 6315 ±44 6159 ±80 3, b 22049 2.148 ±0.029 1.35 5107 ±21 5153 ±42 5049 ±48 5207 ±80 4, c 23249 2.394 ±0.029 1.21 4986 ±57 5150 ±51 5027 ±48 - 5, a 26965 1.504 ±0.006 0.40 5143 ±14 5153 ±38 5098 ±32 5311 ±80 6, d ... 1.650 ±0.060 3.63 4910 ±90 ... ... ... 4, d 128621 6.001 ±0.021 0.35 5182 ±24 5234 ±63 5168 ±75 - 7, e 146233 0.676 ±0.006 0.89 5836 ±46 5818 ±13 5810 ±12 5826 ±80 8, f ... 0.780 ±0.017 2.18 5433 ±69 ... ... ... 9, f 209100 1.890 ±0.020 1.06 4527 ±29 4754 ±89 4649 ±73 4731 ±80 4, g References for θLD: (1) Teixeira et al. (2009); (2) Pijpers et al. (2003); (3) van Belle & von Braun (2009); (4) Kervella & Fouqu´e (2008); (5) Th´evenin et al. (2005); (6) Boyajian et al. (2012b); (7) Kervella et al. (2003); (8) Bazot et al. (2011); (9) Boyajian et al. (2012a). References for fbol: (a) Bruntt et al. (2010); (b) van Belle & von Braun (2009); (c) Cayrel et al. (2011); (d) Boyajian et al. (2012b); (e) Ram´ırez & Mel´endez (2004); (f) Boyajian et al. (2012a); (g) Ram´ırez & Mel´endez (2005). -400 -200 0 200 400 4500 5000 5500 6000 6500 ∆Teff (K) Teff (K) This Work Interferometry - this work -400 -200 0 200 400 ∆Teff (K) Interferometry - SO08 -400 -200 0 200 400 ∆Teff (K) Interferometry - IRFM Figure 3.15: Comparison between the spectroscopic, the IRFM and the direct temperature measurements for stars in common with the literature. The x-axis corresponds to temperatures derived from this work. Square symbols are for the different temperatures for HD 26965, stars for HD 10700 and triangles for HD 146233. Stars with grey color represent Teff with high angular diameter uncertainty. 3.7. NEW ATMOSPHERIC PARAMETERS FOR COOL PLANET HOSTS 61 the spectroscopic and photometric methods. A precision better than 2% in the angular diameters corresponds to an accuracy of 1% in the effective temperatures, which is roughly 60 K at solar temperature, assuming no error in the bolometric flux. It is useful thus, to take the uncertainty of the angular diameter into consideration for a reliable determination of temperature. 3.7 New atmospheric parameters for cool planet hosts The planet-host stars with effective temperature below 5200 K derived using the line list of SO08, imply that they have overestimated temperatures. This has implications for both their mass and radius determination. The lower temperatures imply lower stellar masses as well as lower stellar radii. This means that the derived planetary masses and radii (for transit planet cases) are also lower. As a consequence of the mass reduction, the semi-major axis of the orbits will also be smaller. The expected effects are however small, and no major revisions are expected to occur. With this new line list, we re-derive the stellar parameters for 10 “cool” planet hosts already published in the literature and are not included in the 451 stellar sample of this work. We only consider GK dwarfs with an effective temperature lower than 5200 K whose planets were detected with the radial velocity technique from the CORALIE and HARPS GTO planet search samples. These planet hosts have been previously analyzed with high S/N spectra following the same procedure as this work but with different line lists. In Table 3.7, the fundamental parameters based on the new line list are presented. The sixth column gives the reference of the previously published parameters. To explain the effect on mass of these new parameters more quantitative, we calculate the stellar masses from the Padova interface for both with the original and new parameters. We avoid using the published stellar masses in order to compare uniformly. We find the maximum difference in mass to be 1.5% in absolute units which is negligible compared to the standard mass error. 3.8 Ionization balance vs. Teff for other elements The precise and accurate stellar parameters are, as well, very important for further analyzing stellar chemical abundances. The traditional spectroscopic abundance analysis methods require these parameters as input to compute the atmosphere models, hence the accuracy of the final elemental abundances depends on the accuracy of these input parameters. Different atoms and ions are not equally sensitive to all the stellar parameters. For example, ionized species are more sensitive to gravity variations than neutral species (e.g., Gilli et al. 2006; Neves et al. 2009; Adibekyan et al. 2012c). Recently, Neves et al. (2009) and Adibekyan et al. (2012c) analyzing chemical abundances of the refractory elements in the HARPS sample stars, observed some unexpected trends with effective temperature. Particularly, they detected systematic trends of [X/H] or [X/Fe] with Teff for some elements at low temperatures and found that [Cr i/Cr ii] 62 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS Table 3.7: Updated stellar parameters for previously analyzed planet hosts. Name Teff log g[Fe/H] ξtReference (K) (cm s−2) (dex) (km s−1) BD-082823 4648 ±135 4.33 ±0.32 0.00 ±0.08 0.27 ±0.81 1 HD 3651 5182 ±79 4.30 ±0.16 0.12 ±0.05 0.66 ±0.15 2 HD 13445 5114 ±61 4.55 ±0.13 -0.29 ±0.04 0.66 ±0.15 2 HD 20868 4720 ±91 4.24 ±0.22 0.08 ±0.06 0.47 ±0.31 1 HD 99492 4815 ±184 4.28 ±0.46 0.24 ±0.12 0.50 ±0.56 4 HD 125595 4596 ±235 4.25 ±0.63 0.10 ±0.14 0.14 ±1.41 3 HD 128311 4778 ±75 4.35 ±0.17 -0.03 ±0.02 0.82 ±0.16 2 HD 192263 4906 ±57 4.36 ±0.17 -0.07 ±0.02 0.78 ±0.12 2 HD 215497 5003 ±103 4.26 ±0.26 0.25 ±0.05 0.61 ±0.22 1 HIP 5158 4673 ±175 4.24 ±0.47 0.22 ±0.12 0.34 ±1.09 1 (1) Sousa et al. (2008); (2) Santos et al. (2004); (3) S´egransan et al. (2011); (4) Santos et al. (2005). and [Ti i/Ti ii] abundance ratios gradually increase with decreasing effective temperature when Teff <5000 K. Similar trends for different elements with Teff have been already noted in the literature (see e.g., Valenti & Fischer 2005; Preston et al. 2006; Gilli et al. 2006; Lai et al. 2008; Suda et al. 2011). Different explanations of the mentioned trends are discussed in the literature. The unexpected trends in the low temperature regime may be due to the stronger line blending and may also be connected to either deviations from excitation or ionization equilibrium, or to problems associated with the differential analysis (Neves et al. 2009). A possible explanation for the observed trends with Teff could also be an incorrect T-τrelationship in the adopted model atmospheres (Lai et al. 2008) or NLTE effects (Bodaghee et al. 2003). Summarizing, it can be assumed that the observed trends are probably not an effect of stellar evolution, and uncertainties in atmospheric models are the dominant effect in measurements (see also the discussion in Adibekyan et al. 2012c). In Fig. 3.16, we plot the [Cr i/Cr ii] and [Ti i/Ti ii] abundance ratios derived using the stellar parameters of SO08 and this work as a function of the Teff for stars cooler than 5000 K. This plot is useful to ensure that the ionization equilibrium enforced on the Fe ii lines is acceptable to other elements. As can be seen the slopes of the abundance ratios with new parameters are very gentle. The new slope of [Cr i/Cr ii] per 1000 K is −0.16 ±0.08, whereas the slope with the parameters of SO08 is −0.49 ±0.08. The new slope of [Ti i/Ti ii] is also improved a lot and is −0.18 ±0.06 dex per 1000 K. For comparison the slope of SO08 is −0.39 ±0.08 dex. Although the trend with Teff is weak, there is a shift of about 0.2 dex for the [Ti i/Ti ii] ratio. This shift is difficult to connect to the still possible uncertainties in the stellar parameters and its exact nature still remains to be clarified. Probably, one (or more) of the above mentioned effects can be responsible for that. Unfortunately, in the literature there is no available NLTE calculations for the Ti iand Ti ii lines used in our study, and it is difficult to estimate 3.9. IMPACT OF THIS WORK 63 Figure 3.16: [Cr i/Cr ii] and [Ti i/Ti ii] as a function of effective temperature. The black squares and red asterisks correspond to the abundance ratios derived using the stellar parameters of SO08 and this work, respectively. The solid and dashed lines depict the linear fits of the data. the NLTE effect for the [Ti i/Ti ii] ratio. Summarizing, this independent test shows that the new stellar parameters derived from the iron lines more carefully chosen for cooler stars make the observed [Cr i/Cr ii] and [Ti i/Ti ii] trends with Teff much weaker. 3.9 Impact of this work With the completion of this work (hereafter TS13) we are able to derive precise and accurate parameters for the cooler stars. Our line list has been used for determining stellar parameters for planet-host stars. After almost a decade of gathering high resolution spectra for planet hosts, our team has compiled a catalog with atmospheric parameters derived from a uniform analysis (Santos et al. 2013). This new catalog of stellar parameters for stars with planets (SWEET-Cat), uses the line list of this work for the cooler planet hosts (Fig. 3.17). Another important application of this work is the determination of stellar parameters 64 CHAPTER 3. STELLAR PARAMETERS FOR COOL STARS for evolved stars, and especially the cool giant stars in recent work of Mortier et al. (2013a). The authors after using different line list sets, concluded that the optimal line list was the one of this work for evolved stars (Fig. 3.18). In their work, they investigated the correlation between planet frequency and stellar metallicity. The authors found no metallicity enhancement for red giants with planets with respect to red giants without planets. Finally, Alves et al. (2015; accepted) published a catalog of accurate stellar atmospheric parameters for a sample of 257 K & G field evolved stars that are being surveyed for planets, using precise radial velocity measurements, as part of the CORALIE program to search for planets around giants. The authors compared different line lists and concluded that the line list of TS13 was the optimal for their analysis. The parameters of that work were later used for chemical abundance determinations in the subsequent work of Adibekyan et al. (2015; submitted). 3.9. IMPACT OF THIS WORK 65 Figure 3.17: Comparison between our baseline stellar parameters with those listed in the Extrasolar Planets Encyclopedia RV planet hosts. Green triangles denote the 48 stars whose parameters are presented in Santos et al. (2013). The dotted line represents a 1:1 relation, and the full line a linear fit to the data. Typical error bars are shown on the upper left part of each panel. Taken from Santos et al. (2013). 72 CHAPTER 4. SPECTRAL SYNTHESIS Finally, assuming LTE for small volumes in the photosphere, the monochromatic flux is calculated: Fν= 2π ∞ Z 0 Bν(tν)E2(tν)dtν,(4.19) where Bνis the Planck function. The above equation can be treated in many ways and different spectral synthesis codes have different approaches. The spectral synthesis code of this work solves the RT equation numerically, using a Gaussian quadrature: Fν=X n=1,N Bν(τi)wi,(4.20) where τiand wiare the quadrature nodes and weights respectively, calculated for orthogonal polynomials, for i=1, ..., 10. We have to note that Bνis a function of temperature. The temperature in each depth scale can be obtained by the model atmospheres. Model atmospheres are calculated on a standard optical depth scale (usually at 5000˚ A). The problem can be solved if we convert the τiscale to the one of the model atmosphere (τstd). The conversion is accomplished by solving the first order equation: dτstd dτν =Kλstd (τstd) Kλν(τstd)(4.21) The total absorption coefficients (Kλstd and Kλν) are calculated from the model atmospheres (see next Section). The above equation is solved by SME iteratively with a Feautrier technique (Mihalas 1982). To sum up, the monochromatic flux is derived at the surface of the photosphere, i.e. at τi= 0. Next, the corresponding τstd is calculated for each of the quadrature nodes i, from Equation 4.21. The Planck function is then calculated for the τiand with the temperature that is obtained from the model atmosphere. The Bν(τi)wiis calculated for each increment till the 10th order and summed for the total flux as in Equation 4.20. 4.1.2 Absorption coefficients The transformation from τνto τstd requires the calculation of the absorption coefficients in the atmosphere of the star (Equation 4.21). The values of Kτstd are directly provided from the model atmosphere. The absorption coefficient for other optical depth scales (Kτν) has to be calculated separately. Firstly, we need to define the two processes of absorption: 1) the continuous opacity absorption and 2) the line opacity absorption. Both the continuous opacity absorption coefficient (κτν) and the line opacity absorption coefficient (ατν) are calculated for every optical depth increment and wavelength. 4.1. CALCULATION OF THE SYNTHETIC SPECTRUM 73 4.1.3 Line absorption coefficient The line absorption coefficient is defined as the amount of energy absorbed from a beam of radiation with specific intensity Iν, in a bound-bound transition process. This process involves: natural atomic absorption, pressure broadening, and thermal Doppler broadening. Natural atomic absorption is caused by the interaction of light with dipoles. Due to that, the electrons oscillate as a harmonic oscillator with a damping constant. The shape of the line absorption coefficient for natural broadening per atom, αis described as: αν=e2 mc γrad/4π ∆ν2+ (γrad/4π)2(4.22) The width of the broadening is described by the damping constant, γrad, that is also called radiation damping. The total energy absorbed for a quantum mechanical treatment is expressed: ∞ Z 0 αdν =πe2 mc f, (4.23) where fis the oscillator strength that is different for each transition and is related to the transition probability. Pressure broadening is caused by the interaction between the atoms absorbing light and other particles such as ions, electrons, atoms or molecules. The atomic levels of the transitions of the absorbers are altered due to the perturbers. This distortion is a function of their separation, R. The change in energy induced by collision can be expressed in the form: ∆W= constant/Rn,(4.24) where n depends on the type of interaction. In case the perturbers are charged particles (ions, electrons), the collision process is called Quadratic Stark effect (n= 4). The numerical value of this collision is expressed: log γ4≈19 + 2 3log C4+ log Pe−5 6log T, (4.25) where Peis the electron pressure density and Tthe temperature (Gray 2005). If the perturber is a neutral particle, such as neutral hydrogen that is dominant in cool stars, the perturbations are called van der Waals (n= 6) and is expressed by: log γ6≈20 + 0.4 log C6+ log Pg−0.7 log T, (4.26) where Pgis the gas pressure density (Gray 2005). These approximations often underestimate the value of γ6and in these cases, an enhancement factor is introduced. In this work, we use the Uns¨old factor that equals to 2.5 (Uns¨old 1955). 74 CHAPTER 4. SPECTRAL SYNTHESIS We have to note that the pressure and thermal damping coefficient depends on the optical depth, whereas the radiation damping is constant throughout the atmosphere. Thermal broadening is caused due to the motions of the atoms along the line of sight. These thermal motions are described by the Boltzmann velocity fields and cause Doppler shifts. The energy absorbed from a unit intensity is: ανdν =π1/2e2 mc f1 ∆νD e−(∆ν/∆νD)2dν, (4.27) where νDis the Doppler frequency shift. Since the thermal motions are caused by Boltzmann velocities, υ0, the shift ∆νDbecomes: ∆νD=υ0 cν0=ν0 c2kT m1/2 ,(4.28) with cthe speed of light, and kthe Boltzmann constant. Other small scale motions can produce the same Doppler shifts as thermal motions. These effects are included in the above expression of the absorption coefficient by adding a velocity distribution of dispersion to the ∆νDvalue. This velocity field is called microturbulence, ξt, as we saw in Sect. 2.5, and is described as following: ∆νD=ν0 c (2kT m+ξ2)1/2. To put everything in context, the line absorption coefficient is a result of the following processes: natural broadening, Stark broadening, van der Waals broadening, thermal, and microturbulence broadening. The first three broadening mechanisms show the same dispersion profile, which means that their damping constants can be combined in one profile: γ=γrad +γ4+γ6. Therefore, we can convolve the new dispersion profile with the Gaussian profile of the νDdispersion. The convolution gives: αν=πe2 mc fγ/4π2 ∆ν2+ (γ/4π)2∗1 π1/2∆νD e−(∆ν/∆νD)2 =π1/2e2 mc f ∆νD H(u, α) (4.29) where H(u, α) is the Hjerting function (Hjerting 1938) with u=∆ν ∆νD α=γ 4π 1 ∆νD (4.30) The Hjerting function is similar to a Voigt function, V(u, α) = H(u, α)/(π1/2∆νD), which is most often used. 4.2. CONVOLUTION WITH VELOCITY FIELDS 75 4.1.4 Continuous absorption coefficient The main processes that invoke continuous absorption in a stellar atmosphere are bound-free, free-free transitions and scattering. The main sources of absorption for both transition types are neutral hydrogen, negative hydrogen ion, hydrogen molecule, helium, and other metals (such as Si, Al, Ca, C, N, O). In the visible region of cool stars, the continuous absorption coefficient (κν) is dominated by the absorption of the negative hydrogen ion (H−). The description of κνdue to H−is: κ∝constant T−5/2Pee0.75/kT,(4.31) where Peis the electron pressure. 4.1.5 Disc integration In order to match the flux of the synthetic spectrum with the observed one, additional steps have to be made. A necessary step is to integrate the above specific intensities over the stellar surface to construct the total disc flux. SME divides the stellar disc into annuli of the same intensity and calculates the integration weights for each of them. The weight is just given by the relative area of each annulus, normalized such that the sum of all weights is unity. Each annulus is defined with a different angle, µ1, of in total seven µangles. Each intensity spectrum is convolved with a kernel which describes the distribution of rotational velocities present in the current annulus. 4.2 Convolution with velocity fields The large scale velocity fields that introduce line broadening are macroturbulence (υmac) and the projected rotational velocity (υsin i). In both cases, the υmac and υsin i profiles are convolved with the flux spectrum. Macroturbulence, in contrast to microturbulence, describes the motion in cells that are larger than the unit optical depth. There are several empirical correlations of υmac with other stellar parameters, such as temperature and surface gravity in the literature (e.g., Valenti & Fischer 2005; Doyle et al. 2014). Another broadening mechanism not related with stellar physics, but the spectrograph itself, is the instrumental broadening. In our case, we use a Gaussian profile that is also convolved with the flux spectrum. Instrumental broadening depends on the resolution of the spectrograph. Now the final spectrum is ready to match the observations. 1µis the cos θ, where θis the angle between the outward intensity and the line of sight. 76 CHAPTER 4. SPECTRAL SYNTHESIS 4.3 Best-fit parameters SME includes the minimization procedure to find the best-fit parameters. In the spectral synthesis approach, the stellar parameters one can derive are the following: temperature, surface gravity, overall metallicity, microturbulence, macroturbulence, rotational velocity, radial velocity, and chemical abundances of individual elements. To do so, SME uses the Levenberg-Marquardt algorithm to solve for the least-squares problem: χ2=XObs −Model Unc 21 N,(4.32) where Obs is the observed spectrum, Unc is the uncertainty on the flux of the Obs, and N are the degrees of freedom (N= number of data points minus the free parameters). The Levenberg-Marquardt technique combines the gradient search for searches that approach the minimum from far away and the expansion method as the search converges. 4.4 Stellar parameters with synthesis SME has been widely used in the community for the parameter determination of FGK and M stars. For instance, Valenti & Fischer (2005) analyzed a sample of 1040 solar-type stars of high-resolution spectra, using wavelength intervals as in Fig. 4.2. Additionally, the same code is used in the analysis of large samples for the GES. Apart from large samples, SME is used in the characterization of numerous planet hosts (e.g., P´al et al. 2010; Bakos et al. 2012; Van Eylen et al. 2014). An interesting study of Torres et al. (2012) presents a uniform analysis of transit planet hosts. The stellar parameters are derived using the methodology of Valenti & Fischer (2005). The authors constrain surface gravity with the one derived directly from the transit light curve. A comparison between constrained and unconstrained parameters shows biases that come mainly from strong correlations between the constrained and unconstrained values of Teff, [Fe/H] with log g(see Fig. 4.3). 4.4. STELLAR PARAMETERS WITH SYNTHESIS 77 Figure 4.2: Observed and over-plotted synthetic spectra for some wavelength intervals. Bold horizontal line segments along the bottom axis demarcate spectral segments used to constrain synthetic spectrum fits. Taken from Valenti & Fischer (2005). 78 CHAPTER 4. SPECTRAL SYNTHESIS Figure 4.3: Impact on the temperatures and metallicities of fixing log gto the photometric values, for three different methods. The panels show the differences in the sense ’constrained minus unconstrained’ as a function of the change in log g. Taken from Torres et al. (2012). CHAPTER 5 Stellar parameters for stars with moderate and fast rotation “Waka waka waka waka waka waka waka waka.” Pacman In this Chapter, I describe a new methodology specially designed to treat spectra of stars with moderate and fast rotation. I used the spectral package SME for this analysis and created a complete automatic procedure to derive precise stellar parameters for FGK dwarfs and giants. This work was published in Tsantaki et al. (2014). 5.1 Stars with moderate and fast rotation The high quality stellar spectra obtained from RV planet search programs (e.g., Sousa et al. 2008, 2010), make spectroscopy a powerful tool for deriving the fundamental parameters in absence of more direct measurements. Direct measurements of stellar mass and radius can be derived from detached eclipsing binaries that are often accurate to 1-2%, providing direct log gdeterminations. Direct determinations of temperature are restricted only to stars with measurements of their angular diameter using the interferometric technique. 79 80 CHAPTER 5. PARAMETERS FOR HIGH ROTATORS A typical method of deriving stellar parameters for solar-type stars, as we saw in previous Chapters, is based on the excitation and ionization equilibrium by measuring the EW of iron lines (hereafter the EW method). This method has successfully been applied to RV targets that are restricted to low rotational velocities (υsin i) to increase the precision of the RV technique (Bouchy et al. 2001). High rotational velocities also limit the precision of the EW method. Spectral lines are broadened by rotation and therefore neighboring lines become blended, often unable to be resolved. Even though the EW is preserved, its correct measurement is not yet possible. On the other hand, the transit planet hosts have a wider dispersion in rotational rates when comparing to the slowly rotating FGK hosts observed with the RV technique. For moderate and fast rotating stars, which may be the case of the transit targets, spectral synthesis is required for the parameter determination. This technique yields stellar parameters by fitting the observed spectrum with a synthetic one (e.g., Valenti & Fischer 2005; Malavolta et al. 2014) or with a library of pre-computed synthetic spectra (e.g., Recio-Blanco et al. 2006). In this chapter, we propose a refined approach based on the spectral synthesis technique to derive stellar parameters for slowly rotating stars (Sect. 5.2), yielding results on the same scale with the homogeneous analysis of our previous works (Sect. 5.3). Our method is tested for a sample of moderate-to-high rotators (Sect. 5.4) and also to a number of planet hosts providing new stellar parameters. Their planetary properties are also revised (Sect. 5.5). 5.2 Spectroscopic analysis Due to severe blending, measuring the EW of stars with high rotational velocity is very difficult, if not impossible (see Fig. 5.1). An approximate limit of the rotational velocity where the EW method provides reliable results, is up to υsin i∼12-15 km s−1 (depending on the choice of lines and spectral type). 5.2.1 Line list For an accurate spectral synthesis, atomic and molecular data of all lines in the wavelength intervals where the synthesis is conducted must be as accurate as possible. The choice of intervals for our analysis is based on the line list of iron lines, as described in Tsantaki et al. (2013) (see also Chapter 3). This list is comprised of weak, isolated iron lines, specifically chosen from the extended line list of Sousa et al. (2008) to exclude blended lines that are commonly found in K-type stars. Effective temperatures derived with this line list are in agreement with the IRFM for the whole temperature regime of FGK dwarfs. The spectral window around each iron line is set wide enough to include broadened lines of υsin i∼50 km s−1. Following the Doppler law, such a rotational velocity causes a broadening of ±1˚ A, around a line in the middle of the optical wavelength range (∼5500 ˚ A). 5.2. SPECTROSCOPIC ANALYSIS 81 0.75 0.8 0.85 0.9 0.95 1 6091 6092 6093 6094 6095 Normalized flux Wavelength (Å) υsini = 10 km/s υsini = 15 km/s υsini = 20 km/s υsini = solar Figure 5.1: Solar absorption lines (black), broadened by different rotational profiles: 10 km s−1(blue), 15 km s−1(red), and 20 km s−1(green). Blending at these rates due to rotation makes the accurate measurement of the EW very difficult. The original line list contains 137 Fe iand Fe ii lines where we set intervals of 2˚ A around them. The atomic data for these intervals were obtained from the VALD (Piskunov et al. 1995; Kupka et al. 1999). We extracted atomic data for all the expected transitions for a star with solar atmospheric parameters for our wavelength intervals. We also included lines predicted for a K-type star with Teff = 4400 K. The two line lists that correspond to atomic transitions for the two different spectral types were merged into one after removing duplicates. Molecular data of the most abundant molecules in solar-type stars (C2, CN, OH, and MgH) were also obtained from VALD using the same requests as for the atomic data. From the above intervals we selected the optimal ones according to the following procedure. From the first analyses, we noticed that K-type stars show the highest residuals between the observed and the best-fit synthetic spectrum compared to the F and G spectral types. The main reason is that the spectra of K-type stars include numerous lines, but not all appear in our line list after the requested atomic data queries. Therefore, we discarded lines in the bluer part (below 5000 ˚ A) where lines are more crowded. Lines within overlapping intervals were merged into one. 88 CHAPTER 5. PARAMETERS FOR HIGH ROTATORS Table 5.4: Differences in stellar parameters between this work and the EW method for the 48 sample stars. MAD values correspond to the median average deviation and are indicated in parenthesis. Nis the number of stars used for the comparison. This Work – EW ∆Teff a(±MAD) K ∆ log g(±MAD) dex ∆ [Fe/H] (±MAD) dex N Whole sample –26 ±14 (±55) –0.19 ±0.04 (±0.14) 0.000 ±0.010 (±0.041) 48 F-type –97 ±22 (±68) –0.34 ±0.07 (±0.18) 0.006 ±0.014 (±0.014) 12 G-type 7 ±16 (±36) –0.07 ±0.05 (±0.04) 0.019 ±0.021 (±0.032) 18 K-type –5 ±27 (±32) –0.18 ±0.05 (±0.16) –0.027 ±0.016 (±0.042) 18 aThe standard errors of the mean (σM) are calculated with the following formula: σM=σ √N,σbeing the standard deviation. 5.4. PARAMETERS FOR FAST ROTATORS 89 The effective temperatures derived with the spectral synthesis technique and the EW method are in good agreement. The greatest discrepancies appear for Teff >6000 K, where the effective temperature derived from this work is systematically cooler. The same systematics are also presented in Molenda- ˙ Zakowicz et al. (2013), where the authors compare the EW method with other spectral synthesis techniques but the explanation for these discrepancies is not yet clear. The values of metallicity are in very good agreement between the two methods with zero mean differences and 0.04 dex median average deviation. Surface gravity is a parameter that is the most difficult to constrain with spectroscopy. The comparison of the two methods shows a considerable offset of 0.19 dex, where log gis underestimated compared to the EW method. Interestingly, this offset is smaller for giant stars (∆ log g= 0.07 dex) than for dwarfs (∆ log g= –0.24 dex). To further investigate these differences, we compare the spectroscopic log gwith surface gravity derived with another method that is less model dependent. For 16 dwarf stars in our sample that have a transiting planet, surface gravity can be derived from the analysis of the transit light curve (see also Sect. 5). We compare log gderived from the transit light curve with the spectroscopic log gfrom the EW method (both values are taken from Mortier et al. (2013b)) and this work (see Fig. 5.3). We show that log gfrom the EW analysis is overestimated for low log gvalues and underestimated for high log gvalues. Fortunately, this trend does not affect Teff and [Fe/H], as shown in the recent work of Torres et al. (2012). The same systematics were also found between the log gfrom the EW method and the log gderived with the Hipparcos parallaxes for solar-type stars in Tsantaki et al. (2013) and Bensby et al. (2014). These results imply that log gfrom the EW method using iron lines suffers from biases, but the explanation is not clear. On the other hand, log gderived from this work is in very good agreement with the transit log g, for values lower than 4.5 dex. Stars with log g > 4.5 dex correspond to the cooler stars and are also underestimated. The reason for this underestimation is not yet known, so further investigation is required to understand this behavior. Despite the differences for the log gvalues of mainly the F-type stars, the results listed in Table 5.4 show that for slowly rotating FGK stars, stellar parameters derived from both methods are on the same scale. This means that for the whole sample, the residuals between both methods are small and of the same order of magnitude as the errors of the parameters. 5.4 Spectroscopic parameters for fast rotating FGK stars Testing our method for slow FGK rotators does not necessary imply that it will work for higher υsin iwhere spectral lines are much more broadened and shallower. Our goal is to examine how efficient our method is for stars with moderate-to-high rotation rates. For this purpose, we derived stellar parameters for reference stars of different spectral type and with low υsin i. Second, these stars are convolved with a set of rotational 90 CHAPTER 5. PARAMETERS FOR HIGH ROTATORS −400 −200 0 200 400 4500 5000 5500 6000 6500 7000 This work − EW Teff EW (K) 4500 5000 5500 6000 6500 7000 Teff This work (K) −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 This work − EW [Fe/H] EW (dex) −0.6 −0.4 −0.2 0 0.2 0.4 [Fe/H] This work (dex) −1 −0.5 0 0.5 1 2.5 3 3.5 4 4.5 5 This work − EW logg EW (dex) 2.5 3 3.5 4 4.5 5 logg This work (dex) Figure 5.2: Comparison between the parameters derived using the spectral synthesis method (This Work) and the results of our EW method: temperature (top panel), metallicity (middle panel) and surface gravity (bottom panel). Filled circles represent dwarf stars and asterisks giants. 5.4. PARAMETERS FOR FAST ROTATORS 91 −1 −0.5 0 0.5 1 4500 5000 5500 6000 6500 7000 ∆logg (dex) Teff This work (K) −1 −0.5 0 0.5 1 4.1 4.2 4.3 4.4 4.5 4.6 4.7 ∆logg (dex) logg transit (dex) This work EW method Figure 5.3: Comparison of surface gravity derived from the transit fit with this work and the EW method. ∆ log grepresents ’transit minus this work’ (red circles) and ’transit minus EW method’ (blue squares). profiles using the rotin3 routine as part of the SYNSPEC synthesis code2(Hubeny et al. 1994). As a result, each star has eight different rotational velocities (initial, 5, 10, 15, 20, 25, 30, 40, 50 km s−1). Stellar parameters of all rotational profiles were calculated to investigate how they differ from the non-broadened (unconvolved) star. This test is an indication of how the accuracy of our method is affected by adding a rotational profile. The selected reference stars are two F-type, one G-type, and four K-type stars, and they are presented in boldface in Table C. Probably one star per spectral type would be enough, but we included more Fand K-type stars because they showed higher uncertainties (especially the K-type stars). In Figs. 5.45.6, we show the differences of stellar parameters between the stars with the unconvolved values (original υsin i), and the convolved ones for the eight different rotational velocities. As υsin iincreases, K-type stars show the highest differences in the stellar parameters compared to the non-broadened profile. These deviations for high υsin iare also shown in the error analysis of Sect. 5.2.4. The temperatures of these stars are systematically underestimated with increasing υsin i. On the other hand, the parameters of Fand G-type stars are very close to the ones with low rotation, and no distinct trends are observed with rotation. Even for very high υsin i, temperature and metallicity can be 2http://nova.astro.umd.edu/Synspec43/ 92 CHAPTER 5. PARAMETERS FOR HIGH ROTATORS -40 -20 0 20 40 60 80 100 120 140 0 10 20 30 40 50 60 ∆Teff (K) υsini (km/s) Sun G-type HD 20852 F-type HD 61421 F-type HD 40307 K-type HD 20868 K-type HD 27894 K-type HD 63454 K-type Figure 5.4: Differences in temperature (initial υsin iminus the different rotational profiles) versus υsin i. Each star is represented with different symbol and each spectral type is represented with different colour. Blue for K-type, red for G-type and green F-type. The initial υsin iis different for each star. 5.4. PARAMETERS FOR FAST ROTATORS 93 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 10 20 30 40 50 60 ∆logg (dex) υsini (km/s) Figure 5.5: Differences in surface gravity (initial υsin iminus the different rotational profiles) versus υsin i. Each star is represented with different symbol and each spectral type is represented with different colour. Blue for K-type, red for G-type and green F-type. The initial υsin iis different for each star. 94 CHAPTER 5. PARAMETERS FOR HIGH ROTATORS -0.05 0 0.05 0.1 0.15 0 10 20 30 40 50 60 ∆[Fe/H] (dex) υsini (km/s) Figure 5.6: Differences in metallicity (initial υsin iminus the different rotational profiles) versus υsin i. Each star is represented with different symbol and each spectral type is represented with different colour. Blue for K-type, red for G-type and green F-type. The initial υsin iis different for each star. derived with differences in values of less than 100 K and 0.05 dex respectively. Surface gravity, however, shows large differences that reach up to ∼0.20 dex. The above discrepancies in the parameters in turn affect the stellar mass and radius. To investigate these offsets, we calculate the mass and radius for all the rotational velocities using the calibration of Torres et al. (2010) but corrected for small offsets to match masses derived from isochrone fits by Santos et al. (2013). The results in Fig. 5.7 show that the mass hardly changes as υsin iincreases. The stellar radius, however, is affected in the same manner as surface gravity with greater radius differences. For example, the maximum difference in log g(∼0.20 dex) causes a deviation in radius of 0.39 R. 5.4.1 Application to FGK fast rotators We selected a sample of FGK dwarfs with moderate-to-high υsin i, which have available several estimates of their parameters with υsin iup to 54 km s−1that have spectra available in the public archives of different high resolution instruments (HARPS, FEROS, ELODIE, and CORALIE). The spectra were already processed with their standard pipeline procedures. We corrected for the radial velocity shifts and in cases of multiple observations, the spectra were summed using the IRAF tools, dopcor and scombine. We derived the stellar parameters with the method in this work, and the results and literature values of the sample are presented in methods used: other spectral synthesis 5.4. PARAMETERS FOR FAST ROTATORS 95 -0.15 -0.1 -0.05 0 0.05 0 10 20 30 40 50 60 ∆Mass (solar) υsini (km/s) -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0 10 20 30 40 50 60 ∆Radius (solar) υsini (km/s) Figure 5.7: Differences in stellar mass (top panel) and radius (bottom panel) vs. υsin i. The symbols are the same as in Fig. 5.4. 96 CHAPTER 5. PARAMETERS FOR HIGH ROTATORS Table 5.5: Differences in parameters derived with different methods. Nindicates the number of stars used for the comparison. ∆Teff (K) ∆ log g(dex) ∆ [Fe/H] (dex) N This Work – EW 3±48 –0.11 ±0.07 0.04 ±0.03 11 (MAD = 80) (MAD = 0.24) (MAD = 0.03) This Work – Synthesis 32 ±29 0.03 ±0.05 0.05 ±0.02 29 (MAD = 64) (MAD = 0.15) (MAD = 0.04) This Work – Photometry -12 ±25 0.06 ±0.02 – 18 (MAD = 44) (MAD = 0.04) – techniques, the EW method (up to υsin i∼10 km s−1), and the photometric technique, namely IRFM. The differences between this work and other methods are very small for all parameters. In Fig. 5.8, we plot the comparison between the literature values and our results. Figure 5.9 shows each stellar parameter in dependence of rotational velocity for the different methods and for this work. Even though the mean differences in temperature are close to zero, a slight overestimation of our method appears for high υsin i. Surface gravity shows the lowest dispersion when compared to trigonometric log gfrom all methods. Metallicity is also in agreement, excluding perhaps an outlier (HD 49933). Some examples of spectral fitting are given in Figs. 5.105.11 for two stars with moderate rotation. 5.5 Data and spectroscopic parameters for planet hosts We have identified spectra for ten confirmed planet hosts that show relatively high υsin i, and we were unable to apply our standard EW method for their spectroscopic analysis. We use the procedure of this work to derive their stellar parameters to update the online SWEET-Cat catalog where stellar parameters for FGK and M planet hosts3 are presented (Santos et al. 2013). These stars were observed with high resolution spectrographs (Table 5.6) gathered by our team (these spectra have never been analyzed before) and from the archive (for the NARVAL spectra). Their spectral type varies from F to G. The spectra were reduced with the standard pipelines and are corrected with the standard IRAF tools for the radial velocity shifts and their spectra are added in cases of multiple exposures of individual observed stars. Following the procedure presented in this work, we derived their fundamental parameters, which are included in Figs. 5.8 and 5.9 and presented in Table 5.7. The stellar masses and radii are calculated using the calibration of Torres et al. (2010) with the corrections of Santos et al. (2013). 3https://www.astro.up.pt/resources/sweet-cat/ 5.5. PARAMETERS FOR PLANET HOSTS 97 -400 -300 -200 -100 0 100 200 300 400 5000 5500 6000 6500 7000 ∆Teff (K) Teff This work (K) 5000 5500 6000 6500 7000 Teff (K) IRFM Synth EW -1 -0.5 0 0.5 1 3.6 3.8 4 4.2 4.4 4.6 4.8 5 ∆logg (dex) logg This work (dex) 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 logg (dex) Hipparchos Synth EW -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 ∆[Fe/H] (dex) [Fe/H] This work (dex) -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 [Fe/H] (dex) Synth EW Figure 5.8: Temperature (top panel), surface gravity (middle panel), and metallicity (bottom panel). Different colors represent different techniques. Square symbols represent planet hosts analyzed in this work. In the middle panel, the average error is plotted. In each panel, the upper plot compares the data and the lower plot compares the residual differences from perfect agreement. 104 CHAPTER 5. PARAMETERS FOR HIGH ROTATORS −10 −5 0 5 10 0.06 0.07 0.08 0.09 0.1 0.11 0.12 ∆(Rp/Rstar)/Rp/Rstar % Rp/Rstar Literature −20 −10 0 10 20 0 5 10 15 20 25 ∆Mp / Mp % Mp (MJ) Literature −40 −20 0 20 40 1 1.1 1.2 1.3 1.4 1.5 1.6 ∆Rp / Rp % Rp (RJ) Literature Figure 5.13: Comparison between the literature data of planetary mass, the radii ratio (Rp/Rstar), and planetary radius and this work, respectively in absolute units. metallicity which consequently propagates to biases in stellar (and planetary) mass and radius. From the planet hosts in our work, there are eight stars with transit data and available log gfrom a light curve analysis. We therefore compare the log gderived from our spectroscopic analysis with the log gfrom the transit fits as taken from the literature (Fig. 5.12). The differences of this comparison are very small (∆ log g= –0.04 with σ= 0.07 dex). On the other hand, a comparison between the log gfrom the transit light curve and the log gusing only the unconstrained methodology of Valenti & Fischer (2005) shows an average difference of 0.18 (σ= 0.27) dex for five stars with available measurements. For completeness, we also plot the log gfrom our light curve analysis of the previous section, using the stellar density and mass. Even though the number of stars for this comparison is very small, these results suggest that fixing log gto the transit value is not required with the analysis of this work, avoiding the biases that are described in Torres et al. (2012). The different approach we adopt in this work, mainly because of the different line list, shows that we obtain a better estimate on surface gravity. However, since our sample is small and limited to hotter stars, further investigation is advised to check whether following the unconstrained approach is the optimal strategy. The unconstrained analysis is also suggested in G´omez Maqueo Chew et al. (2013) as preferable, after analyzing the transit host WASP-13 with SME but following different methodology (line list, initial parameters, convergence criteria, fixed parameters) from Valenti & Fischer (2005). We explored how the literature values of planetary mass and radius are affected with 5.5. PARAMETERS FOR PLANET HOSTS 105 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 ρstellar (g cm−3) This work ρstellar (g cm−3) Literature Figure 5.14: Comparison between stellar density derived from the transit light curve analysis and literature data. the new stellar parameters. From our analysis we find that the dispersion between the planetary mass derived with our stellar parameters and the literature is 4% (Fig. 5.13, top panel). The planet-to-star radius ratio derived from the transit light curve shows the same dispersion of 4% (Fig. 5.13, middle panel). This consistency with the literature values confirms the accuracy of the transit light curve analysis for deriving the planetto-star radius ratio. The planetary radius is calculated from this ratio and the stellar radius that is inferred from our spectroscopic values. The comparison of the planetary radius with the literature values shows the highest dispersion of 14% (Fig. 5.13, bottom panel). Since we have shown the consistency of the planet-to-star radius ratio, the main source of uncertainty in the derivation of planetary radius is the calculation of the stellar value. We also compare the stellar density derived from the transit analysis with the respective ones from the literature (Fig. 5.14). In Fig. 5.15, we show the new mass and radius from this work in comparison with the literature values. Planetary radius shows higher discrepancies mainly because of the uncertainties in the stellar radius calculations. The study of planet hosts with higher rotational velocities is essential because they expand the planet sample around stars of earlier types (Fand A-type) that are more massive than the Sun. Precise stellar parameters for these stars are necessary to study the frequency of planets around intermediate mass stars and explore their planet formation mechanisms. Additionally, precise (and if possible accurate) stellar parameters are essential for a detailed characterization of the planets to be discovered by the upcoming high precision transit missions such as CHEOPS, TESS, and PLATO 2.0. 106 CHAPTER 5. PARAMETERS FOR HIGH ROTATORS 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1 1 10 Rp (RJ) Mp (MJ) Figure 5.15: Blue squares represent planetary mass and radius derived in this work in comparison with literature values (green circles). Characteristic isodensity curves are plotted for the densities of Saturn (dashed), Jupiter (solid) and Neptune (dotted). CHAPTER 6 Conclusions and future prospects “I’ll see you on the dark side of the moon” Pink Floyd 6.1 Conclusions This thesis was divided into two main projects, both related to optimizing the methods of deriving precise and accurate stellar parameters. Our work can be summarized as: •We used the EW method and the existing tools by our team (e.g. ARES) to derive stellar parameters for 451 FGK stars. We fixed the discrepancies between our method and other more model-independent methods for the case of cool stars (Teff <5000 K). These discrepancies were mainly due to the strong blending effects in the spectra of cool stars. After we carefully defined a line list with isolated lines, we achieved better parameters in the low temperature regime. •The stellar parameters for stars with rotation cannot be derived using the standard EW method. For these stars we provided new methodology with the spectral 107 108 CHAPTER 6. CONCLUSIONS AND FUTURE PROSPECTS synthesis technique, based on the code SME in an automatic analysis. We used our experience on line lists to create the most effective wavelength intervals to deal with fast rotating stars. •For this project, we used observational data to determine the stellar parameters of planet hosts with moderate and fast rotation. Their parameters were added to the online catalog of planet-host stars. 6.2 Future work Our methods are proven to be powerful tools to analyze stars of spectral types from F to K. However, there are some issues we would like to address for future work. •It has been shown that EW method systematically overestimates temperature only in the cases of hot stars when compared to other methods (Fig. 6.1). In the future, we aim to solve this problem by designing a line list specially for F-type stars, following the same procedure as for cool stars. Another possible reason for this disagreement could be because of the atomic data. As mentioned in Chapter 2, the atomic data are derived after an inverted solar analysis. Therefore, we expect stars far from the solar parameters to be affected by not precise values in the atomic data. •Surface gravity is a parameter that is the most difficult to constrain with spectroscopy. In our recent work, we compared surface gravities from the EW method and the spectral synthesis technique with the one derived from the transit fit. The transit surface gravity is described as more reliable compared to the spectroscopic. From our analysis, we show that the difference log gtransit – log gEW correlates with temperature while the differences of our spectral synthesis values appear only for the cool stars. Mortier et al. (2013b) made the same analysis by comparing log gtransit with log gEW , and reported the same results (Fig. 6.2). An interesting study is to understand why the EW method gives log gsystematically higher for cool stars and lower for hotter. •We used the spectral synthesis tools to derive the fundamental stellar parameters for stars with moderate-to-high rotation. However, there is more information to obtain from the spectra, i.e. to measure the chemical abundances. The chemical content of a star is important. We can use our experience of iron lines to create intervals around the most important elements. A complete line list to start with is the one of Neves et al. (2009) that included 12 elements (Si, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Na, Mg, and Al) but it can also be expanded to volatile elements. •SME is a powerful package that includes all the procedures for an automatic parameter derivation. However, there is a variety of procedures in the literature that we would like to test using exactly the same methodology. An interesting 6.2. FUTURE WORK 109 Figure 6.1: Top panel: Comparison between the effective temperature derived from Sousa et al. (2010) (This work) and Casagrande et al. (2010). Bottom panel: Comparison between the effective temperature derived from Sousa et al. (2010) (This work) and the measurements of other authors. The crosses, triangles, and diamonds represent the comparison points with the values determined by Valenti & Fischer (2005), Bensby et al. (2003), and Edvardsson et al. (1993), respectively. Taken from Sousa et al. (2010). 110 CHAPTER 6. CONCLUSIONS AND FUTURE PROSPECTS Figure 6.2: Differences in log g(defined as ’photometric - spectroscopic’) as a function of the effective temperature. Taken from Mortier et al. (2013b). test would be the comparison of different spectral synthesis codes, such as iSpec (Blanco-Cuaresma et al. 2014). In addition, we can check our procedure with the line list provided by the GES (early results are presented in Appendix D). The GES is making an great effort to optimize the line lists with precise atomic data. The use of the GES line list will follow the purpose of homogenizing stellar parameters from different working groups for the huge sample of stars observed for the survey. •There is still a large amount of data from follow-up observations for the characterization of planet hosts. I am participating as PI and Co-I in ongoing observing missions at ESO facilities. Once the stellar parameters are derived, we plan to address the question how stellar parameters correlate with the planetary properties. One interesting project is to study the connection of planet hosts with the Galactic parameters, such as the galactocentric distance of the stars (e.g. Adibekyan et al. 2014). Appendices 111 APPENDIX A The microturbulence relationship Microturbulence is taken into consideration for abundance analyses to reconcile differences between the observed and predicted from models EWs of strong lines. Previous studies of FGK dwarfs have shown that ξtdepends on Teff and log g(e.g., Nissen 1981; Reddy et al. 2003; Allende Prieto et al. 2004; Adibekyan et al. 2012a; Ram´ırez et al. 2013). Using a linear regression analysis to the new parameters of the sample, we derive the following expression: ξt= 6.932(±0.125) ×10−4Teff −0.348(±0.042) log g−1.437(±0.182).(A.1) Here, ξtis in km s−1,Teff and log gare in their traditional units. The parameters of the stars in the sample range: 4400 < Teff <6400 K, 3.6 <log g < 4.8 dex, –0.8 <[Fe/H] <0.4 dex. The new derived parameters indicate a linear dependence on temperature for a set value of surface gravity. In Fig. A.1, we see the dependence of microturbulence on temperature for a set of log gvalues. Microturbulence clearly increases with temperature and decreases with surface gravity. 113 120 APPENDIX B. SME TUTORIAL APPENDIX C Stellar parameters with synthesis 121 122 APPENDIX C. STELLAR PARAMETERS WITH SYNTHESIS Table C.1: Results of the comparison between this work and the EW method for dwarf stars. The stars in boldface are analyzed in Sect. 5.3. This work EW method Star Teff log g[Fe/H] υsin i Teff log g[Fe/H] (K) (dex) (dex) (km s−1) (K) (dex) (dex) CoRoT-2 5620 ±18 4.66 ±0.06 -0.03 ±0.03 9.97 5697 ±97 4.73 ±0.17 -0.09 ±0.07 CoRoT-10 4921 ±25 4.09 ±0.09 0.15 ±0.03 2.19 5025 ±155 4.47 ±0.31 0.06 ±0.09 CoRoT-4 6164 ±30 4.34 ±0.11 0.15 ±0.03 7.03 6344 ±93 4.82 ±0.11 0.15 ±0.06 CoRoT-5 6254 ±30 4.41 ±0.11 0.04 ±0.03 1.43 6240 ±70 4.46 ±0.11 0.04 ±0.05 HD 101930 5083 ±18 4.15 ±0.06 0.10 ±0.03 0.10 5083 ±63 4.35 ±0.13 0.16 ±0.04 HD 102365 5588 ±18 4.07 ±0.06 -0.30 ±0.03 0.10 5616 ±41 4.40 ±0.06 -0.28 ±0.03 HD 103774 6582 ±30 4.47 ±0.11 0.27 ±0.03 8.93 6732 ±56 4.81 ±0.06 0.29 ±0.03 HD 1237 5588 ±18 4.58 ±0.06 0.11 ±0.03 4.62 5489 ±40 4.46 ±0.11 0.06 ±0.03 HD 134060 5914 ±18 4.28 ±0.06 0.09 ±0.03 1.44 5940 ±18 4.42 ±0.03 0.12 ±0.01 HD 1388 5967 ±18 4.38 ±0.06 0.00 ±0.03 1.27 5970 ±15 4.42 ±0.05 0.00 ±0.01 HD 148156 6212 ±30 4.40 ±0.11 0.23 ±0.03 5.73 6251 ±25 4.51 ±0.05 0.25 ±0.02 HD 162020 4798 ±25 4.14 ±0.09 -0.14 ±0.03 1.46 4723 ±71 4.31 ±0.18 -0.10 ±0.03 HD 20852 6675 ±30 4.12 ±0.11 -0.37 ±0.03 7.06 6813 ±92 4.76 ±0.12 -0.35 ±0.06 HD 20868 4745 ±25 4.02 ±0.09 0.00 ±0.03 0.46 4720 ±91 4.24 ±0.47 0.08 ±0.01 HD 221287 6337 ±30 4.43 ±0.06 0.02 ±0.06 3.92 6417 ±25 4.60 ±0.10 0.06 ±0.02 HD 222237 4618 ±25 3.92 ±0.09 -0.50 ±0.03 0.10 4722 ±55 4.34 ±0.15 -0.39 ±0.06 HD 23079 5965 ±18 4.28 ±0.06 -0.13 ±0.03 0.10 6009 ±14 4.50 ±0.05 -0.11 ±0.01 HD 27894 4894 ±25 4.08 ±0.09 0.18 ±0.03 0.87 4833 ±209 4.30 ±0.48 0.26 ±0.10 HD 31527 5915 ±18 4.40 ±0.06 -0.17 ±0.03 2.36 5917 ±13 4.47 ±0.05 -0.17 ±0.01 HD 330075 4924 ±30 4.03 ±0.09 -0.04 ±0.03 0.10 4958 ±52 4.24 ±0.13 0.05 ±0.03 HD 361 5924 ±18 4.48 ±0.06 -0.10 ±0.03 0.10 5888 ±14 4.54 ±0.08 -0.13 ±0.01 HD 38283 5962 ±18 4.14 ±0.06 -0.15 ±0.03 4.51 5980 ±24 4.27 ±0.03 -0.14 ±0.02 HD 40307 4771 ±25 4.10 ±0.09 -0.42 ±0.03 0.10 4774 ±77 4.42 ±0.16 -0.36 ±0.02 HD 61421 6616 ±30 4.09 ±0.11 0.03 ±0.03 4.40 6612 4.02 -0.02 HD 63454 4833 ±25 4.11 ±0.09 0.04 ±0.03 1.81 4756 ±77 4.32 ±0.22 0.13 ±0.05 HD 750 5118 ±18 4.34 ±0.06 -0.29 ±0.03 0.10 5069 ±32 4.33 ±0.1 -0.30 ±0.02 HD 870 5379 ±18 4.36 ±0.06 -0.12 ±0.03 0.10 5360 ±24 4.40 ±0.08 -0.12 ±0.02 HD 93385 5987 ±18 4.38 ±0.06 0.02 ±0.03 1.06 5989 ±17 4.46 ±0.03 0.03 ±0.01 HD 967 5643 ±18 4.38 ±0.06 -0.59 ±0.03 0.10 5595 ±18 4.59 ±0.02 -0.66 ±0.01 OGLE-TR-113 4793 ±25 4.25 ±0.09 0.05 ±0.03 5.02 4781 ±166 4.31 ±0.41 0.03 ±0.06 WASP-29 4782 ±25 4.13 ±0.09 0.18 ±0.03 0.10 5203 ±102 4.93 ±0.21 0.17 ±0.05 WASP-15 6378 ±30 4.24 ±0.11 0.03 ±0.03 5.13 6573 ±70 4.79 ±0.08 0.09 ±0.03 WASP-16 5710 ±18 4.23 ±0.06 0.12 ±0.03 0.47 5726 ±22 4.34 ±0.05 0.13 ±0.02 WASP-17 6666 ±30 4.26 ±0.06 -0.04 ±0.03 9.93 6794 ±83 4.83 ±0.09 -0.12 ±0.05 WASP-2 5105 ±18 3.97 ±0.06 0.08 ±0.03 2.90 5109 ±72 4.33 ±0.14 0.02 ±0.05 WASP-23 5053 ±18 4.20 ±0.06 -0.02 ±0.03 0.46 5046 ±99 4.33 ±0.18 0.05 ±0.06 WASP-38 6247 ±30 4.25 ±0.11 0.06 ±0.03 8.05 6436 ±60 4.80 ±0.07 0.06 ±0.04 WASP-6 5447 ±18 4.42 ±0.06 -0.11 ±0.03 0.10 5383 ±41 4.52 ±0.06 -0.14 ±0.03 Sun 5771 ±18 4.42 ±0.06 0.00 ±0.03 2.57 – – – The values of the Sun were calculated from observations of the reflected light from Ganymede with S/N of 150. 123 Table C.2: Results of the comparison between this work and the EW method for giant stars. The stars in boldface are analyzed in Sect. 5.3. This work EW method Star Teff log g[Fe/H] υsin i Teff log g[Fe/H] (K) (dex) (dex) (km s−1) (K) (dex) (dex) HD 148427 5018 ±25 3.49 ±0.09 0.01 ±0.03 0.45 4962 ±45 3.39 ±0.12 0.03 ±0.03 HD 175541 5097 ±18 3.44 ±0.06 -0.14 ±0.03 2.45 5111 ±38 3.56 ±0.08 -0.11 ±0.03 HD 27442 4852 ±25 3.48 ±0.09 0.23 ±0.03 2.65 4781 ±76 3.46 ±0.19 0.33 ±0.05 HD 62509 5007 ±25 3.06 ±0.09 0.21 ±0.03 3.76 4935 ±49 2.91 ±0.13 0.09 ±0.04 HD 88133 5330 ±18 3.62 ±0.06 0.20 ±0.03 3.39 5438 ±34 3.94 ±0.11 0.33 ±0.05 HD 142091 4898 ±25 3.24 ±0.09 0.05 ±0.03 4.38 4876 ±46 3.15 ±0.14 0.13 ±0.03 HD 188310 4799 ±18 3.14 ±0.06 -0.06 ±0.03 5.28 4714 ±49 2.53 ±0.11 -0.27 ±0.04 HD 163917 5107 ±18 2.82 ±0.06 0.33 ±0.03 4.21 4967 ±61 2.70 ±0.13 0.14 ±0.05 124 APPENDIX C. STELLAR PARAMETERS WITH SYNTHESIS Table C.3: Stellar parameters for a sample of fast rotating FGK dwarfs. Star TeffIRF M Ref. log gHIP Teff log g[F e/H] Ref. TeffSynth log gSynth [Fe/H]Synth Ref. TeffEW log gEW [Fe/H]EW Ref. υsin i K dex K dex dex K dex dex K dex dex km s−1 HD 179949 6205 ±80 (1) 4.38 ±0.10 6237 ±30 4.40 ±0.11 0.17 ±0.03 This work 6168 ±44 4.34 ±0.06 0.11 ±0.03 (3) 6287 ±28 4.54 ±0.04 0.21 ±0.02 (18) 6.52 HD 165185 5932 ±80 (1) 4.47 ±0.10 5940 ±18 4.46 ±0.06 -0.05 ±0.03 This work 5906 4.44 -0.07 (4) 5942 ±85 4.53 ±0.13 0.02 ±0.10 19 7.53 HAT-P-6 – – – 6933 ±30 4.38 ±0.11 -0.02 ±0.03 This work 6353 ±88 3.84 ±0.12 -0.23 ±0.08 (5) 6855 ±111 4.69 ±0.20 -0.08 ±0.11 (20) 8.06 HAT-P-23 – – – 5924 ±30 4.28 ±0.11 0.16 ±0.03 This work 5905 ±80 4.48 ±0.12 0.15 ±0.04 (10) – – – – 8.50 HD 19994 6159 ±80 (1) 4.10 ±0.10 6145 ±30 4.10 ±0.11 0.20 ±0.03 This work 6188 ±44 4.24 ±0.06 0.17 ±0.03 (3) 6289 ±46 4.48 ±0.05 0.24 ±0.03 (18) 8.51 HD 89744 6262 ±92 (1) 3.97 ±0.10 6300 ±30 4.07 ±0.11 0.24 ±0.03 This work 6291 ±44 4.07 ±0.06 0.20 ±0.03 (3) 6234 ±45 3.98 ±0.05 0.22 ±0.05 (21) 8.86 HD 49933 6609 ±80 (1) 4.40 ±0.10 6904 ±44 4.18 ±0.15 -0.17 ±0.04 This work 6780 ±70 4.30 ±0.20 -0.30 ±0.11 (6) 6522 4.00 -0.49 22 10.14 HD 142 6313 ±80 (1) 4.27 ±0.10 6271 ±44 4.17 ±0.15 0.13 ±0.04 This work 6249 ±44 4.19 ±0.06 0.08 ±0.03 (3) 6403 ±65 4.62 ±0.07 0.09 ±0.05 (18) 10.22 HD 142860 6336 ±80 (1) 4.27 ±0.10 6361 ±44 4.07 ±0.15 -0.09 ±0.04 This work 6262 ±44 4.18 ±0.06 -0.14 ±0.03 (3) 6281 4.06 -0.13 (22) 10.65 HD 89569 6439 ±80 (1) 4.12 ±0.10 6469 ±44 4.08 ±0.15 0.09 ±0.04 This work 6401 3.99 -0.12 (4) – – – – 11.33 HD 86264 6381 ±80 (1) 4.16 ±0.10 6300 ±44 4.06 ±0.15 0.25 ±0.04 This work 6326 ±44 4.22 ±0.05 0.16 ±0.03 (3) 6596 ±78 4.47 ±0.15 0.37 ±0.06 (23) 12.55 HD 121370 6141 ±80 (1) 3.83 ±0.10 6080 ±44 3.78 ±0.15 0.33 ±0.04 This work 6030 ±80 3.90 ±0.08 0.24 ±0.07 (7) 6300 4.18 0.29 (22) 13.10 Kepler-410A 6273 ±140 (2) – 6375 ±44 4.25 ±0.15 0.09 ±0.04 This work 6325 ±75 – 0.01 ±0.10 8 – – – – 13.24 HD 210302 6477 ±80 (1) 4.29 ±0.10 6405 ±44 4.24 ±0.15 0.10 ±0.04 This work 6339 ±44 4.15 ±0.06 0.08 ±0.03 (3) – – – – 13.68 HD 105 6035 ±80 (1) 4.46 ±0.10 6045 ±44 4.40 ±0.15 0.02 ±0.04 This work 6126 ±44 4.65 ±0.06 -0.02 ±0.03 (3) 6012 ±68 4.42 ±0.12 0.06 ±0.07 (24) 14.43 HD 202917 5579 ±80 (1) 4.57 ±0.10 5539 ±10 4.58 ±0.06 0.03 ±0.01 This work 5617 ±44 4.39 ±0.06 0.03 ±0.03 (3) 5592 ±79 4.31 ±0.17 -0.04 ±0.08 (24) 14.75 WASP-3 – – – 6423 ±44 4.42 ±0.15 0.04 ±0.04 This work 6400 ±100 4.25 ±0.05 0.00 ±0.20 (9) 6448 ±123 4.49 ±0.08 -0.02 ±0.08 (25) 15.21 HD 30652 6499 ±80 (1) 4.33 ±0.10 6494 ±44 4.29 ±0.15 0.04 ±0.04 This work 6424 ±44 4.07 ±0.06 0.00 ±0.03 (3) – – – – 17.01 CoRoT-3 – – – 6558 ±44 4.25 ±0.15 0.14 ±0.04 This work 6740 ±140 4.22 ±0.07 -0.02 ±0.06 (11) – – – – 18.46 XO-3 – – – 6781 ±44 4.23 ±0.15 -0.08 ±0.04 This work 6429 ±50 3.95 ±0.06 -0.20 ±0.02 (12) – – – – 18.77 HAT-P-41 – – – 6479 ±51 4.39 ±0.22 0.13 ±0.05 This work 6390 ±100 3.68 ±0.06 0.21 ±0.10 (13) – – – – 20.11 HAT-P-2 – – 4.22 ±0.10 6414 ±51 4.18 ±0.22 0.04 ±0.05 This work 6290 ±60 4.16 ±0.03 0.14 ±0.08 (14) – – – – 20.50 HAT-P-34 – – – 6509 ±51 4.24 ±0.22 0.08 ±0.05 This work 6442 ±88 3.98 ±0.10 0.22 ±0.04 (15) – – – – 24.08 HD 8673 – – 4.29 ±0.10 6472 ±51 4.27 ±0.22 0.14 ±0.05 This work 6340 ±44 4.21 ±0.06 0.07 ±0.03 (3) – – – – 26.91 HD 82558 – – 4.63 ±0.11 4934 ±70 4.50 ±0.21 -0.14 ±0.11 This work 5062 ±44 5.12 ±0.06 -0.21 ±0.03 (3) – – – – 26.97 CoRoT-11 – – – 6343 ±72 4.27 ±0.30 0.04 ±0.03 This work 6440 ±120 4.22 ±0.23 -0.03 ±0.08 (16) – – – – 36.72 HD 64685 6907 ±80 (1) 4.15 ±0.12 6702 ±98 3.97 ±0.20 -0.14 ±0.08 This work 6995 4.36 0.00 (4) – – – – 41.59 30 Ari B 6396 ±80 (1) 4.39 ±0.12 6284 ±60 4.35 ±0.25 0.12 ±0.06 This work 6314 ±55 4.29 ±0.07 0.11 ±0.04 (17) – – – – 42.61 HD 219877 6741 ±80 (1) 4.11 ±0.13 6620 ±137 4.10 ±0.18 -0.01 ±0.06 This work 6775 4.06 -0.13 (4) – – – – 54.00 (1) Casagrande et al. (2011); (2) Van Eylen et al. (2014); (3) Valenti & Fischer (2005); (4) Gray et al. (2006); (5) Noyes et al. (2008); (6) Bruntt et al. (2004); (7) Bruntt et al. (2010); (8) Huber et al. (2013); (9) Pollacco et al. (2008); (10) Bakos et al. (2011); (11) Deleuil et al. (2008); (12) Johns-Krull et al. (2008); (13) Hartman et al. (2012); (14) P´al et al. (2010); (15) Bakos et al. (2012); (16) Gandolfi et al. (2010) (17) Prugniel et al. (2011); (18) Sousa et al. (2008); (19) Santos et al. (2005); (20) Ammler-von Eiff et al. (2009); (21) Santos et al. (2004); (22) Takeda (2007); (23) Santos et al. (2013); (24) Viana Almeida et al. (2009); (25) Montalto et al. (2012) APPENDIX D The Gaia-ESO line list Gaia-ESO (GES) is a public spectroscopic survey, targeting 105stars, systematically covering all major components of the Milky Way, from halo to star forming regions, providing the first homogeneous overview of the distributions of kinematics and elemental abundances. One of the key goals of this survey is to provide homogeneous parameters for the the large amount of the observed spectra from the ESO facilities. We used the GES line list (version 4) to derive stellar parameters for the GES benchmark stars presented in Jofr´e et al. (2014). The GES methogology suggests to use MARCS models and the provided line list. I used the high resolution and high S/N spectra provided in their database for the benchmark stars. I used the same methodology described in Chapter 5. The only difference in this analysis is the use of the line list and model atmospheres. I kept the same wavelength intervals but included only the lines with their atomic data provided but the GES. The results for the stars that this method is applied are shown below. The extremely metal poor stars and M-dwarfs were excluded. 125 126 APPENDIX D. THE GAIA-ESO LINE LIST 4000 4500 5000 5500 6000 6500 7000 4000 4500 5000 5500 6000 6500 7000 Teff (K) Teff Benchmarks (K) GES linelist TS14 -400 -200 0 200 400 ∆ Teff (K) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 [Fe/H] [Fe/H] Benchmarks GES linelist TS14 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 ∆ [Fe/H] 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 logg logg Benchmarks GES linelist TS14 -1.5 -1 -0.5 0 0.5 1 1.5 ∆ logg Figure D.1: Comparison between the stellar parameters derived with our methodology (green points) and our methodology+the GES line list (red points). Top figure: The x-axis shows the literature values for temperature as published in Jofr´e et al. (2014). Middle figure: Same for metallicity. Bottom figure: Same for surface gravity. APPENDIX E The iron line list Table E.1: The complete line list used for the spectroscopic analysis of Chapter 3 with the atomic parameters of Fe iand Fe ii as well as the corresponding EWs of the Sun λ(˚ A) χlog gf Element EW (m˚ A) 4523.40 3.65 –1.871 FeI 44.2 4537.67 3.27 –2.870 FeI 17.4 4551.65 3.94 –1.928 FeI 29.1 4556.93 3.25 –2.644 FeI 26.3 4566.52 3.30 –2.156 FeI 46.2 4574.22 3.21 –2.353 FeI 41.0 4593.53 3.94 –1.921 FeI 29.5 4596.41 3.65 –2.090 FeI 34.1 4602.00 1.61 –3.163 FeI 72.2 4630.12 2.28 –2.488 FeI 74.3 4631.49 4.55 –1.890 FeI 11.6 4661.54 4.56 –1.186 FeI 38.5 4690.14 3.69 –1.550 FeI 58.8 4802.88 3.69 –1.527 FeI 60.4 4808.15 3.25 –2.630 FeI 27.7 4809.94 3.57 –2.542 FeI 19.4 4811.05 3.07 –3.182 FeI 14.5 4885.43 3.88 –1.136 FeI 72.5 4961.92 3.63 –2.301 FeI 26.7 5127.36 0.92 –3.317 FeI 99.4 5141.74 2.42 –2.125 FeI 89.3 5223.19 3.63 –2.252 FeI 29.4 5228.38 4.22 –1.095 FeI 60.0 5242.50 3.63 –1.124 FeI 86.7 5243.78 4.26 –1.022 FeI 62.2 5247.06 0.09 –4.941 FeI 66.4 5294.55 3.64 –2.627 FeI 15.5 5295.32 4.42 –1.518 FeI 29.3 5376.83 4.29 –2.040 FeI 14.7 5379.58 3.69 –1.552 FeI 61.2 5386.34 4.15 –1.709 FeI 32.1 5389.48 4.42 –0.534 FeI 84.6 5398.28 4.45 –0.684 FeI 73.3 5409.14 4.37 –1.051 FeI 55.7 5432.95 4.45 –0.729 FeI 70.6 5436.30 4.39 –1.319 FeI 40.5 127 128 APPENDIX E. THE IRON LINE LIST 5464.28 4.14 –1.595 FeI 38.2 5466.99 3.65 –2.141 FeI 34.2 5473.17 4.19 –1.986 FeI 19.5 5522.45 4.21 –1.419 FeI 44.0 5543.94 4.22 –1.070 FeI 62.2 5546.51 4.37 –1.124 FeI 52.0 5560.22 4.43 –1.064 FeI 52.4 5584.77 3.57 –2.189 FeI 35.8 5618.64 4.21 –1.298 FeI 50.5 5619.60 4.39 –1.435 FeI 35.0 5633.95 4.99 –0.385 FeI 66.5 5636.70 3.64 –2.511 FeI 19.8 5638.27 4.22 –0.809 FeI 77.6 5649.99 5.10 –0.785 FeI 36.2 5651.47 4.47 –1.763 FeI 18.4 5653.87 4.39 –1.402 FeI 36.7 5679.03 4.65 –0.756 FeI 59.5 5680.24 4.19 –2.330 FeI 10.3 5715.09 4.28 –0.847 FeI 72.5 5720.90 4.55 –1.805 FeI 14.9 5738.24 4.22 –2.164 FeI 13.6 5775.08 4.22 –1.124 FeI 59.8 5793.92 4.22 –1.622 FeI 34.0 5811.92 4.14 –2.333 FeI 11.4 5814.81 4.28 –1.820 FeI 23.0 5815.22 4.15 –2.364 FeI 10.5 5853.15 1.49 –5.130 FeI 7.5 5855.08 4.61 –1.531 FeI 22.3 5862.36 4.55 –0.404 FeI 87.6 5902.48 4.59 –1.797 FeI 14.2 5905.68 4.65 –0.775 FeI 58.7 5927.79 4.65 –1.057 FeI 42.9 5929.68 4.55 –1.211 FeI 39.5 5930.19 4.65 –0.326 FeI 87.9 5934.66 3.93 –1.091 FeI 76.4 5956.70 0.86 –4.526 FeI 53.7 5983.69 4.55 –0.719 FeI 67.2 5987.07 4.79 –0.478 FeI 70.2 6005.55 2.59 –3.479 FeI 22.4 6024.06 4.55 –0.124 FeI 110.5 6056.01 4.73 –0.489 FeI 72.6 6078.49 4.79 –0.364 FeI 77.9 6079.01 4.65 –1.008 FeI 45.8 6089.57 4.58 –1.273 FeI 35.3 6094.38 4.65 –1.566 FeI 19.9 6096.67 3.98 –1.776 FeI 38.2 6120.25 0.92 –5.894 FeI 5.2 6127.91 4.14 –1.417 FeI 48.9 6151.62 2.18 –3.298 FeI 49.7 6157.73 4.08 –1.238 FeI 61.5 6159.38 4.61 –1.878 FeI 11.9 6173.34 2.22 –2.877 FeI 68.0 6200.32 2.61 –2.397 FeI 73.0 6219.29 2.20 –2.463 FeI 89.6 6226.74 3.88 –2.069 FeI 29.2 6232.65 3.65 –1.240 FeI 83.3 129 6240.65 2.22 –3.292 FeI 48.3 6270.23 2.86 –2.573 FeI 52.8 6315.81 4.08 –1.645 FeI 40.5 6322.69 2.59 –2.368 FeI 76.0 6335.34 2.20 –2.339 FeI 97.2 6358.68 0.86 –3.907 FeI 84.4 6392.54 2.28 –3.942 FeI 17.8 6481.88 2.28 –2.929 FeI 64.1 6593.88 2.43 –2.384 FeI 84.6 6609.12 2.56 –2.632 FeI 65.7 6627.55 4.55 –1.475 FeI 28.2 6646.94 2.61 –3.915 FeI 10.3 6699.15 4.59 –2.106 FeI 8.2 6705.11 4.61 –1.057 FeI 46.5 6710.32 1.49 –4.810 FeI 15.9 6713.74 4.79 –1.425 FeI 21.2 6725.36 4.10 –2.187 FeI 17.6 6726.67 4.61 –1.045 FeI 47.2 6732.07 4.58 –2.144 FeI 7.7 6739.52 1.56 –4.902 FeI 11.7 6745.97 4.08 –2.657 FeI 7.2 6839.84 2.56 –3.377 FeI 29.9 6842.69 4.64 –1.169 FeI 39.3 6855.72 4.61 –1.674 FeI 18.6 6857.25 4.08 –2.075 FeI 22.4 6858.15 4.61 –0.972 FeI 51.6 6861.94 2.42 –3.795 FeI 18.9 6864.32 4.56 –2.229 FeI 6.8 4508.28 2.86 –2.403 FeII 87.3 4520.22 2.81 –2.563 FeII 81.9 4576.34 2.84 –2.947 FeII 64.9 4656.98 2.89 –3.676 FeII 33.8 4731.47 2.89 –2.515 FeII 82.1 4923.93 2.89 –1.541 FeII 154.3 5197.57 3.23 –2.293 FeII 80.1 5234.63 3.22 –2.235 FeII 83.5 5264.81 3.23 –3.091 FeII 45.6 5337.75 3.23 –3.338 FeII 35.5 5414.07 3.22 –3.568 FeII 26.9 5991.38 3.15 –3.539 FeII 31.2 6149.25 3.89 –2.719 FeII 36.2 6247.56 3.89 –2.347 FeII 52.2 6442.97 5.55 –2.399 FeII 5.1 6456.39 3.90 –2.110 FeII 63.0 6516.09 2.89 –3.279 FeII 54.2 136 BIBLIOGRAPHY Bond, J. 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