Improved fundamental parameters and LTE abundances of the CoRoT ††thanks: The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESA’s RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain solar-type pulsator HD 49933
Key Words.:Stars: abundances – Stars: fundamental parameters – Stars: individual: HD 49933
Aims:The knowledge of accurate stellar parameters is a key stone in several fields of stellar astrophysics, such as asteroseismology and stellar evolution. Although the parameters can be derived both via spectroscopy and multicolor photometry, the obtained results are sometimes affected by systematic uncertainties. In this paper we present a self-consistent spectral analysis of the solar-type star HD 49933, which is a primary target for the CoRoT satellite.
Methods:We used high-resolution and high signal-to-noise ratio spectra to carry out a consistent parameter estimation and abundance analysis of HD 49933. The LLmodels code was employed for model atmosphere calculations, while SYNTH3 and WIDTH9 codes were used for line profile calculation and LTE abundance analysis.
Results:In this paper we provide a detailed description of the methodology adopted to derive the fundamental parameters and the abundances. Although the obtained parameters differ from the ones previously derived by other authors, we show that only the set obtained in this work is able to fit the observed spectrum accurately. In particular, the surface gravity was adjusted to fit pressure-sensitive spectral features.
Conclusions:We confirm the importance of a consistent analysis of relevant spectroscopic features, application of advanced model atmospheres, and the use of up-to-date atomic line data for the determination of stellar parameters. These results are crucial for further studies, e.g. detailed theoretical modelling of the observed pulsation frequencies.
In the last ten years several space missions were launched to obtain high quality photometric data (e.g. WIRE, MOST, CoRoT) and several others are going to be launched in the near future (e.g. KEPLER, BRITE). One of the main goals of these missions is to provide astronomers with high precision photometric data that will allow a better understanding of stellar pulsation – the only approach that can improve our knowledge about stellar interiors.
The modelling of pulsational signals requires the knowledge of stellar parameters and primarily accurate values of the effective temperature () and metallicity (). The determination of fundamental parameters can be performed by different methods (some examples for stars from B- to G-type are: Fuhrmann et al. 1997; Przybilla et al. 2006; Fossati et al. 2009) that not always lead to consistent results. Thus it is important to choose a methodology that allows to constrain the parameters of the star from the available observables (usually photometry and spectroscopy) in the most robust and reliable way.
After Mosser et al. (2005) discovered the presence of solar-like oscillations in HD 49933 using RV measurements from high resolution time-series spectra, this star was included as a primary CoRoT target for the photometric observation of solar-like oscillations. It was one of the first main sequence solar-type star in which solar-like oscillations were clearly detected from space photometry, thanks to the high quality data provided by the CoRoT satellite (Appourchaux et al. 2008).
Fundamental parameters and abundances of HD 49933 were obtained previously by different authors with different methods. The published fundamental parameters of HD 49933 display some scatter both in and (e.g. Edvardsson et al. 1993; Blackwell & Lynas-Gray 1998; Gillon & Magain 2006; Cenarro et al. 2007; Bruntt et al. 2008). As a consequence, the published abundances show some scatter. At the same time a need for a higher precision in stellar parameters for HD 49933 was emphasised by recent asteroseismic modelling of this object (Appourchaux et al. 2008; Kallinger et al. 2009), which employed new pulsation analysis techniques (e.g. Gruberbauer et al. 2008).
The main goal of the present work is to perform a consistent atmospheric and abundance analysis of HD 49933 which fits all photometric and spectroscopic data and to investigate to what degree the derived fundamental parameters depend on the applied methods.
2 Observations and spectral reduction
Ten spectra of HD 49933 were obtained between February 11th and February 13th
2006 with the cross-dispersed echelle spectrograph HARPS (spectral
resolution 115 000) attached to the 3.6-m ESO La Silla
telescope. The spectra were reduced with the online pipeline111http://www.eso.org/sci/facilities/lasilla/
Each spectrum was obtained with an exposure time between 243 and 300 seconds. We retrieved the spectra from the ESO archive and co-added them to obtain a single spectrum with a signal-to-noise ratio (SNR) per pixel of about 500, calculated at 5000 Å. The final spectrum, normalised by fitting a low order spline to carefully selected continuum points, covers the wavelength range 3780–6910 Å. The spectrum has a gap between 5300 Å and 5330 Å, since one echelle order is lost in the gap between the two chips of the CCD mosaic detector.
Figure 4 (Online material) shows (from top to bottom) a sample of the first spectrum obtained on February 11th, the last obtained on February 13th and the final co-added spectrum, around the strong Fe ii line at 5018 Å.
3 The model atmosphere
To compute model atmospheres of HD 49933 we employed the LLmodels stellar model atmosphere code (Shulyak et al. 2004). For all the calculations Local Thermodynamical Equilibrium (LTE) and plane-parallel geometry were assumed. We used the VALD database (Piskunov et al. 1995; Kupka et al. 1999; Ryabchikova et al. 1999) as a source of atomic line parameters for opacity calculations. For a given model atmosphere we performed a line selection procedure which allowed to chose lines that significantly contribute to the opacity for a given set of model parameters, adopting the selection threshold %, where and are the continuum and line absorption coefficients at a given frequency . Convection was implemented according to the Canuto & Mazzitelli (1991a, b) model of convection (see Heiter et al. 2002, for more details).
4 Fundamental parameters and abundance analysis
Gillon & Magain (2006) used Strömgren indices to determine the atmospheric parameters of HD 49933. By using the TEMPLOGG code (Rogers 1995) they found = 6543200 K, = 4.240.20 and dex. Although the uncertainties are quite large, we used this as our starting point and in an iterative process to gradually improve the parameters using different spectroscopic indicators as we will describe in detail below. In our analysis, every time any of , , or abundances changed during the iteration process, we recalculated a new model with the implementation of the last measured quantities. The same concerns newly derived abundances: while the results of the abundance analysis depend upon the assumed model atmosphere, the atmospheric temperature-pressure structure itself depends upon the adopted abundances, so that we recalculated the model atmosphere every time abundances were changed, even if the other model parameters were fixed. Such a procedure ensures the model structure to be consistent with the assumed abundances.
We performed the determination by fitting synthetic line profiles, calculated with SYNTH3 (Kochukhov 2007), to the observed profiles of three hydrogen lines: H, H and H. In the temperature range expected for HD 49933 hydrogen lines are extremely sensitive to temperature variations and very little to variations, therefore being good temperature indicators. The obtained with this procedure is = 6500 50 K.
Figure 1 shows the comparison between the observed H line profile and the synthetic profiles calculated with the adopted stellar parameters, as well as the synthetic profile obtained with the higher = 6780 130 K published by Bruntt et al. (2008).
Another spectroscopic indicator for is given by the analysis of metallic lines. In particular, is determined eliminating the correlation between line abundance and the line excitation potential () for a given ion/element. This procedure can lead to erroneous parameters, so we decided not to take into account this indicator in our analysis. This important point will be extensively discussed in Sect. 5.
The surface gravity was derived from two independent methods based on line profile fitting of Mg i lines with developed wings and ionisation balance for several elements. The first method is described in Fuhrmann et al. (1997) and is based on the fact that the wings of the Mg i lines at 5167, 5172 and 5183 Å are very sensitive to variations. In practice we derived first the Mg abundance from other Mg i lines without developed wings, such as 4571, 4730, 5528 and 5711 Å, and then we fit the wings of the other three lines by tuning the value. To apply this method, it is necessary to have very accurate log values and Van der Waals (log ) damping constants for all the lines. The log values of the Mg lines given in the VALD database are of rather high quality and came from the laboratory measurements of the lifetimes (Anderson et al. 1967). However, recently new laboratory measurements of the oscillator strengths for Mg i lines at 5167, 5172 and 5183 Å with an accuracy log 0.04 dex were published by Aldenius et al. (2007). Van der Waals damping constants in VALD are calculated by Barklem & O’Mara (2000) but they appear to be slightly higher than needed to fit the solar lines. While log in Barklem & O’Mara (2000), Fuhrmann et al. (1997) derived log from the fitting of the solar lines using the Anderson et al. (1967) oscillator strengths. For our analysis we employed different combinations of oscillator strengths and damping constants. Adopting oscillator strengths from Aldenius et al. (2007) and damping constants from Barklem & O’Mara (2000) or oscillator strengths from Anderson et al. (1967) and damping constants from Fuhrmann et al. (1997) we derived a value of 3.85, while adopting oscillator strengths from Aldenius et al. (2007) and damping constants from Fuhrmann et al. (1997) we obtained a value of 4.00. We obtained a formal value of 3.93 0.07 where the given error bar on mainly depends on the uncertainty of the damping constant (). Comparison of the synthetic colors with the observed ones (see Sect. 5.2) favors a value of . Figure 2 shows a comparison between the synthetic and observed Mg i line profile for the 5172 and 5183 Å Mg i lines. The same procedure applied to the solar flux spectrum (Kurucz et al. 1984) resulted in =4.44 (oscillator strengths from Aldenius et al. 2007 and damping constants from Fuhrmann et al. 1997) and =4.30 for the combination of the oscillator strengths from Aldenius et al. (2007) and damping constants from Barklem & O’Mara (2000). The comparison of the observed and the synthetic Mg i line profile for the 5172 and 5183 Å Mg i lines for the Sun is shown in Fig. 2.
For surface gravity determination one often uses ionisation equilibrium, but this method is extremely sensitive to the non-LTE effects present for each ion/element, while Mg lines with developed wings (less sensitive to non-LTE effects) are more suitable as indicators (Zhao & Gehren 2000). For this reason we decided to keep the Mg lines as our primary indicator checking afterwards the obtained result with the ionisation equilibrium. The value we obtained (=4.000.15) is lower than that given by photometry and by other authors, e.g. Gillon & Magain (2006): =4.260.08 and Bruntt et al. (2008): =4.240.13. The difference between our result and, for example, the one of Gillon & Magain (2006); Bruntt et al. (2008) is clearly connected with the difference in the derived effective temperature, while there is a reasonable agreement for , but the error bars are unfortunately quite large. This important point will be discussed in Sect. 5.
Since HD 49933 does not have an effective temperature high enough to show He lines, we are not able to measure the atmospheric He abundance. Diffusion calculations for low-mass metal poor stars show that helium should be depleted in the atmospheric layers (Proffitt & Michaud 1991), therefore we tested the effect of a helium underabundance on the obtained parameters. We rederived and respectively with hydrogen lines and Mg i line profiles assuming a helium abundance of dex and of dex. In both cases we obtained no visible change of the hydrogen line profiles indicating no effect on the determination, while should be increased of about 0.15 with both helium underabundances, to be able to fit again the Mg i line profiles adopted to derive . This value is within the adopted uncertainty for .
Our main source for the atomic parameters of spectral lines is the VALD database. LTE abundance analysis was based on equivalent widths, analysed with a modified version (Tsymbal 1996) of the WIDTH9 code (Kurucz 1993a). Although 301 Fe i lines were measured, only 158 lines with accurate experimental oscillator strengths were chosen for abundance determination to achieve the highest accuracy. First, we reject the lines with theoretical oscillator strengths. The rest of the lines were checked in the solar flux spectrum (Kurucz et al. 1984), observed with a resolving power at wavelengths between 4000 Å and 4700 Å and with at longer wavelengths. The final choice was made for the lines that did not require an oscillator strength correction greater than 0.1 dex to fit the line cores of the solar spectrum. We also tried to have a set of Fe i lines uniformly distributed over the range of equivalent widths and excitation potentials. As for the lines of other elements/ions we used nearly all unblended spectral lines with accurate atomic parameters available in the wavelength range 3850–6880 Å, except lines in spectral regions where the continuum normalisation was too uncertain. In case of blended lines, for lines subjected to hyperfine splitting (hfs) or for lines situated in the wings of the hydrogen lines we derived the line abundance performing synthetic spectrum calculations with the SYNTH3 code. The hfs constants for abundance calculations were taken from Blackwell-Whitehead et al. (2005) for Mn i lines, from Biehl (1976) for Ba ii 4554 Å line and from Lawler et al. (2001) for Eu ii lines. hfs effects are negligible for the two Cu i lines we used in the abundance analysis.
|Ion||HD 49933 - this paper||Bruntt et al. (2008)||Sun|
|6500 K||6780 K||5777 K|
The projected rotational velocity and macroturbulence () were determined fitting synthetic spectra of several carefully selected lines in the observed spectrum. We obtained = 10 0.5 and = 5.2 0.5 . The value derived for is in agreement with that expected according to the relation - published by Valenti & Fisher (2005) and obtained on a sample of more than a thousand stars.
Mosser et al. (2005) detected a line profile distortion variable with time with an amplitude of less than 500 , probably due to granulation. This distortion is too small to be able to affect the parameter and abundance determination since the parameters were mainly derived with hydrogen and Mg i line profiles, while abundances used equivalent widths. The distortion of the line profiles could have, at most, increased the microturbulence velocity. It is known that Am stars show also distorted line profiles that lead to an increase in of 1.5-2.0 , but for Am stars the distortion is of the order of 3-4 (Landstreet 1998). If granulation played a role in increasing the derived value of , it is probably less than 0.1 , which is below our detection limit.
The microturbulence was determined by minimising the correlation between equivalent width and abundance for several ions. We mainly used Fe i lines since this is the ion that provides the largest number of measured lines within a wide range in equivalent width, but also the correlations obtained with Ti i, Ti ii, Cr i, Fe ii, and Ni i lines were taken into account. The range of the microturbulent velocities goes from 1.4 (Ti i) to 1.9 (Ti ii) with an average of =1.60 0.18 . The microturbulent velocity derived from Fe i lines is 1.5 . This value, with an uncertainty of , was adopted for the final analysis as the best representation for all lines of neutral elements. Figure 5 (Online material) displays the correlation between the line abundance of Fe i and the measured equivalent widths.
The final abundances are given in Table 1. We also computed the abundance difference between HD 49933 and the solar atmosphere as derived by Asplund et al. (2005) (4th column) and by Grevesse & Sauval (1998) (5th column). The results of the abundance analysis of HD 49933 taken from Bruntt et al. (2008) are given in the 6th column of Table 1 for comparison.
The stellar metallicity () is defined as follows:
where is the atomic number of an element with atomic mass m. Making use of the abundances obtained from the performed analysis we derived a metallicity of = 0.008 0.002 dex, adopting the solar abundances by Asplund et al. (2005) for all the elements that were not analysed. However, if we assume an underabundance of dex for all not analysed elements the resulting value remains practically unchanged. Scaling the solar abundances of all elements by dex gives the metallicity = 0.006 dex. This substantial difference in illustrates that it is important to have an accurate determination of C and O, which make a large contribution to the determination of . Additionally, a He underabundance of and dex relative to Sun does not change the value.
4.1 Abundance uncertainties
The abundance uncertainties shown in Table 1 are the standard deviation of the mean abundance from the individual line abundances and do not represent the real error bars associated with each derived element abundance.
Based on realistic errors on the equivalent width measurements we find that the uncertainty on the abundances of a given line is only about 0.01 dex (see Fossati et al. 2009, for more details). In the case of ions with a sufficiently high number of lines we assume that the internal scatter for each ion includes the uncertainties due to equivalent width measurement and continuum normalisation.
Plotting the abundance scatter as a function of the number of lines it is possible to conclude about the mean scatter that can be applied in cases when only one line of a certain ion is measured. In these cases it is reasonable to assume an internal error of 0.10 dex.
The internal scatter is just a part of the total abundance error bar since the uncertainties on the fundamental parameters are also playing an important role. Table 3 (Online material) shows the variation in abundance resulting from the changing of each parameter (, and ) by and the final adopted uncertainty for each ion following the standard error analysis.
The main source of the uncertainty is the effective temperature, while the uncertainties due to and in particular to are almost negligible.
Assuming that the contributions to the uncertainty on the abundances are independent, we derive the final uncertainty using error propagation. The result is given in column seven of Table 3 (available as Online material).
5.1 Comparison with previous determinations
Our abundance calculation with the use of different atmospheric models for HD 49933 proposed in literature shows that the accuracy (rms) of the abundance determination is the same for practically all of them and any difference in absolute value is caused by the differences in the model parameters. Therefore we shall compare and discuss here the determination of the atmospheric parameters.
The published effective temperature values range from 6300 K (Hartmann & Gehren 1988) through 6484 K (Blackwell & Lynas-Gray 1998) (IR flux method), 6595 K (Edvardsson et al. 1993) (Strömgren photometry) to 6735–6780 K (Gillon & Magain 2006; Bruntt et al. 2004, 2008). The last three cited works employed spectrum synthesis iterative methods: APASS, VWA. For more details on previously derived atmospheric parameters see also Table 4 of Bruntt et al. (2004). The APASS and VWA methods use the correlation of the abundance of neutral Fe lines with excitation potential as an indicator of . The final is found by minimising this correlation. Hydrogen lines are not considered in APASS and VWA methods. It is important to emphasise that in the APASS method (Gillon & Magain 2006) the continuum level is adjusted in each iteration. Gillon & Magain (2006) also obtained very different temperature estimates working with APASS (=673553 K) and with equivalent widths (=653844 K). The authors gave a favor for the APASS results motivating that the Gaussian approximation of the line profiles (used to measure equivalent widths) in spectrum of a star rotating with =10 overestimates the equivalent widths. Since we used equivalent widths we investigated this possible error source by calculating synthetic profiles of 26 Fe i lines in the range 6–122 mÅ with our adopted model atmosphere and with Fe abundance of , measured their equivalent widths with direct integration, broadened the synthetic spectrum with the instrumental, rotational and macroturbulent velocity profiles, and then again measured equivalent widths by Gaussian approximation of the convolved spectrum. Inserting both sets of equivalent widths in the WIDTH9 code we got an average abundance of adopting directly integrated equivalent widths and of adopting the Gaussian approximation, as shown in Table 4 present in the Online material. No significant dependencies indicating any change of microturbulent velocity or effective temperature appeared in the second case. Thus we conclude that in HD 49933 the use of equivalent widths instead of synthetic spectrum should not influence the determination of the model parameters.
The use of both the APASS and the VWA (Bruntt et al. 2008) methods also leads to a higher for HD 49933 and another star HD 32115, for which the effective temperature was derived by Bikmaev et al. (2002) with the same method (Strömgren photometry and hydrogen line profiles) as in the present paper. Figure 3 shows individual abundances from 158 Fe i lines calculated with the model parameters derived in this paper ( = 6500 K, = 4.00) and by Bruntt et al. (2008) ( = 6780 K, = 4.24) versus the excitation energy of the lower level. For both models the microturbulent velocity, derived by us in the usual way, is nearly the same, 1.5 and 1.6 , respectively. However, to be consistent with the results by Bruntt et al. (2008) we plot abundances versus excitation potential for Bruntt’s model using =1.8 derived in Bruntt et al. (2008). The same dependence was also calculated for HD 32115 using equivalent width measurements from Bikmaev et al. (2002) and model atmospheres from Bikmaev et al. (2002) and from Bruntt et al. (2008), respectively. As a final check we used the solar flux spectrum. We analysed 137 out of 158 Fe i lines with the solar model atmosphere ( = 5777 K, = 4.44) also calculated with the LLmodels code. There were 21 discarded lines that could not be properly fitted in the solar spectrum. Seven lines are blended in the Sun which is cooler and more metallic than HD 49933, while the rest 14 lines have extended wings and cannot be accurately measured. Our analysis resulted in for =0.95 in full agreement with the currently adopted solar parameters (Asplund et al. 2005). Individual abundances derived from the solar Fe i lines versus the excitation energy of the lower level are shown on Figure 3.
In Fig. 3 one immediately notices the negative slope, indicating a too high , adopting the model parameters obtained by (Bruntt et al. 2008) for both stars. This fact together with the poor fit of the hydrogen line profiles (see Fig. 1) clearly shows that both automatic methods based on synthetic spectrum calculations, ignoring hydrogen lines, lead to an overestimate of the effective temperature by 200 K. With our model we get small positive slope that formally corresponds to an underestimate of the by 30 K. For the Sun zero slope is obtained.
Our temperature is close to that derived with the IR flux method by Blackwell & Lynas-Gray (1998), =648445 K, and provides a good description of all global observables: photometric colors, hydrogen and metallic line profiles.
Our value for the surface gravity is supported by the ionisation equilibrium for Fe i/Fe ii and few other elements, such as Si i/Si ii and Ca i/Ca ii. Even for Ti and Cr, for which we have enough lines for an accurate abundance analysis, the average abundances from the two ionisation stages agree within the error bars. Applying a higher microturbulence of 1.9 , derived from Ti ii lines, for example, leads to a Ti i/Ti ii equilibrium. In the case of chromium about half of the analysed Cr ii lines and one third of Cr i lines have theoretically calculated oscillator strengths which may influence the final abundance results. The atmospheric parameters derived for HD 49933 are similar to another solar-type pulsator, Procyon, which has =651249 K, =3.960.02, and (Allende Prieto et al. 2002). The comparison of the observed and the synthetic Mg i line profile for the 5172 and 5183 Å Mg i lines for Procyon is shown in Fig. 2 (see Ryabchikova et al. 2008, for details concerning the observed spectrum of Procyon and the adopted model atmosphere).
5.2 Synthetic photometry
Since there are no available observations of HD 49933 in the visible spectral region, calibrated to physical units, we adopted photometric observations extracted from the SIMBAD database222http://simbad.u-strasbg.fr/simbad/. Table 2 summarises the comparison between observed and theoretical color-indexes of different photometric systems. The theoretical colors were computed using computer codes by Kurucz (1993a) modified to read and process high-resolution fluxes produced by the LLmodels. The reddening, corresponding to , was derived using analytical extinction models by Amôres & Lépine (2005). Besides the final model with K, (Model 1), we present also the synthetic photometry for three other models: K, (Model 2); a model with the parameters derived from the Strömgren photometry ( K, , Model 3), and a model with the parameters published by Bruntt et al. (2008) ( K, , Model 4).
|index||(Model 1)||(Model 2)||(Model 3)||(Model 4)|
It is seen from the Table 2 that the Strömgren and UBV indexes are better described by Model 1. The H index shows a poor fit for all models, but this does not play a critical role in the present study since the profiles of hydrogen lines were fitted perfectly with the parameters of Model 1. The color-indexes of the Geneva system are generally better described by Model 1,2 and 3 i.e. models with lower . However, the indexes and are better fitted with a higher temperature (Model 4). Model 1 and 3 show a comparable fit, with the difference that Model 1 fits better the sensitive index, but worse the index, while for Model 3 it is the opposite. In summary, we find that the majority of the photometric indicators and the present accurate spectroscopic study point to the adopted K and =4.0.
Based on ten high-resolution, high signal-to-noise spectra taken from ESO archive we carried out a precise spectroscopic analysis of one of the primary solar-like CoRoT targets HD 49933. All ten spectra were averaged to obtain a single spectrum with a resolution and signal-to-noise ratio per pixel at Å. Using this spectrum we revised the fundamental parameters and the atmospheric abundances in a consistent way employing modern 1D LTE stellar model atmospheres (LLmodels; Shulyak et al. 2004). The derived set of fundamental parameters and abundances provide a good fit to the available observables: IR and multicolor photometry, pressure-sensitive magnesium lines, metallic lines and profiles of hydrogen Balmer lines.
It is shown that the implementation of automatic procedures for the abundance analysis using only line spectra without hydrogen lines can result in inaccurate parameters and hence abundances.
Acknowledgements.This work was supported by following funding projects: FWF Lise Meitner grant Nr. M998-N16 (DS), by the Presidium RAS Programme “Origin and evolution of stars and galaxies”, and by the Leading Scientific School grant 4354.2008.2 (TR). LF has received support from the Austrian Science Foundation (FWF project P19962 - Modulated RR Lyrae Stars). We kindly thank Thomas Kallinger and Michael Gruberbauer for the useful discussions and comments during the preparation of the draft.
- Aldenius et al. (2007) Aldenius, M., Tanner, J.D., Johansson, S., Lundberg, H., & Ryan, S.G. 2007, A&A, 461, 767
- Allende Prieto et al. (2002) Allende Prieto, C., Asplund, M., Garćia Lopez, R.J., & Lambert, D. 2002, ApJ, 567, 544
- Amôres & Lépine (2005) Amôres, E. B., Lépine, J. R. 2005, AJ, 130, 659
- Anderson et al. (1967) Anderson, E.M., Zilitis, V.A., & Sorokina, E.S. 1967, Opt. Spectr., 23, 102
- Appourchaux et al. (2008) Appourchaux, T., Michel, E., Auvergne, M. et al. 2008, A&A, 488, 705
- Asplund et al. (2005) Asplund, M., Grevesse, N., & Sauval, A. J. 2005, Astronomical Society of the Pacific Conference Series, 336, 25
- Barklem & O’Mara (2000) Barklem,P.S., & O’Mara, B.J. 2002, MNRAS, 311, 535
- Bikmaev et al. (2002) Bikmaev, I. F., Ryabchikova, T.A., Bruntt, H., et al. 2002, A&A, 389, 537
- Biehl (1976) Biehl, D. 1976, PhD thesis, Christian-Albrechts-Universität, Kiel, Institute für Theoretische
- Blackwell & Lynas-Gray (1998) Blackwell, D. E., & Lynas-Gray, A. E. 1998, A&AS, 129, 505
- Blackwell-Whitehead et al. (2005) Blackwell-Whitehead, R. J., Pickering, J. C., & Pearse, O. 2005, ApJS, 157, 402
- Bruntt et al. (2004) Bruntt, H., Bikmaev, I. F., Catala, C., et al. 2004, A&A, 425, 683
- Bruntt et al. (2008) Bruntt, H., De Cat, P., & Aerts, C. 2008, A&A, 478, 487
- Burleigh et al. (1997) Burleigh, M.R., Barstow, M.A., & Fleming, T.A. 1997, 1997, MNRAS, 287, 381
- Canuto & Mazzitelli (1991a) Canuto, V. M., & Mazzitelli, I. 1991, ApJ, 370, 295
- Canuto & Mazzitelli (1991b) Canuto, V. M., & Mazzitelli, I. 1992, ApJ, 389, 724
- Cenarro et al. (2007) Cenarro, A. J., Peletier, R. F., Sánchez-Blázquez, P., et al. 2007, MNRAS, 374, 664
- Edvardsson et al. (1993) Edvardsson, B., Andersen, J., Gustafsson, B., et al. 1993, A&A, 275, 101
- Fossati et al. (2009) Fossati, L., Ryabchikova, T., Bagnulo, S., et al. 2009, A&A, accepted
- Fuhrmann et al. (1997) Fuhrmann, K., Pfeiffer, M., Frank, C., Reetz, J., & Gehren, T. 1997, A&A, 323, 909
- Gillon & Magain (2006) Gillon, M., & Magain, P. 2006, A&A, 448, 341
- Grevesse & Sauval (1998) Grevesse, N., & Sauval, A. J. 1996, Astronomical Society of the Pacific Conference Series, 99, 117
- Gruberbauer et al. (2008) Gruberbauer, M., Kallinger, T., & Weiss, W. W. 2008, A&A, submitted, arXiv: 0811.3345
- Guenther et al. (1992) Guenther, D. B., Demarque, P., Kim, Y.-C., & Pinsonneault, M. H. 1992, ApJ, 387, 372
- Hartmann & Gehren (1988) Hartmann, K., & Gehren, T. 1988, A&A, 199, 269
- Heiter et al. (2002) Heiter, U., Kupka, F., van’t Veer–Menneret, C., Barban, C., et al. 2002, A&A, 392, 619
- Kallinger et al. (2009) Kallinger, T., Gruberbauer, M., Guenther, D. B., Fossati, L., & Weiss, W. W. 2009, A&A, accepted; arXiv: 0811.4686
- Kochukhov (2007) Kochukhov, O. 2007, Spectrum synthesis for magnetic, chemically stratified stellar atmospheres, Physics of Magnetic Stars, 109, 118
- Kupka et al. (1999) Kupka, F., Piskunov, N., Ryabchikova, T. A., Stempels, H. C., & Weiss, W. W. 1999, A&AS, 138, 119
- Kurucz (1993a) Kurucz, R. 1993, ATLAS9: Stellar Atmosphere Programs and 2 km/s grid. Kurucz CD-ROM No. 13 (Cambridge: Smithsonian Astrophysical Observatory)
- Kurucz et al. (1984) Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, NSO Atlas No. 1: Solar Flux Atlas from 296 to 1300 nm, Sunspot, NSO
- Landstreet (1998) Landstreet, J. D. 1998, A&A, 338, 1041
- Lawler et al. (2001) Lawler, J. E., Wickliffe, M. E., Den Hartog, E. A., & Sneden, C. 2001b, ApJ, 563, 1075
- Marigo et al. (2008) Marigo, P., Girardi, L., Bressan, A., et al. 2008, A&A, 482, 883
- Mosser et al. (2005) Mosser, B., Bouchy, F., Catala, C., et al. 2005, A&A, 431, L13
- Pietrinferni et al. (2004) Pietrinferni, A., Cassisi, S., Salaris, M., & Castelli, F. 2004, ApJ, 612, 168
- Piskunov et al. (1995) Piskunov, N. E., Kupka, F., Ryabchikova, T. A., Weiss, W. W., & Jeffery, C. S. 1995, A&AS, 112, 525
- Proffitt & Michaud (1991) Proffitt, C. R. & Michaud, G. 1991, ApJ, 371, 584
- Przybilla et al. (2006) Przybilla, N., Butler, K., Becker, S. R., & Kudritzki, R. P. 2006, A&A, 445, 1099
- Rogers (1995) Rogers, N. Y. 1995, Comm. Astroseismol., 78
- Ryabchikova et al. (1999) Ryabchikova, T. A., Piskunov, N. E., Stempels, H. C., Kupka, F., & Weiss, W. W. 1999, Phis. Scr., T83, 162
- Ryabchikova et al. (2008) Ryabchikova, T. A., Kochukhov, O., & Bagnulo, S. 2008, A&A, 480, 811
- Shulyak et al. (2004) Shulyak, D., Tsymbal, V., Ryabchikova, T., Stütz Ch., & Weiss, W. W. 2004, A&A, 428, 993
- Valenti & Fisher (2005) Valenti, J. A., & Fisher, D. A. 2005, ApJS, 159, 141
- Tsymbal (1996) Tsymbal, V. V. 1996, in ASP Conf. Ser. 108, Model Atmospheres and Spectral Synthesis, ed. S. J., Adelman, F., Kupka, & W. W., Weiss, 198
- Zhao & Gehren (2000) Zhao, G., & Gehren, T. 2000, A&A, 362, 1077
|Direct integration||Gaussian measurements|