Spectroscopic Determination of Atmospheric Parameters of Solar-Type Stars: Description of the Method and Application to the Sun

Transcription

1 PASJ: Publ. Astron. Soc. Japan 54, , 2002 June 25 c Astronomical Society of Japan. Spectroscopic Determination of Atmospheric Parameters of Solar-Type Stars: Description of the Method and Application to the Sun Yoichi TAKEDA Komazawa University, Komazawa, Setagaya, Tokyo takedayi@cc.nao.ac.jp Michiko OHKUBO and Kozo SADAKANE Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara-shi, Osaka okubo@cosmos2.cc.osaka-kyoiku.ac.jp, sadakane@cc.osaka-kyoiku.ac.jp (Received 2002 March 4; accepted 2002 April 23) Abstract A computer program has been developed for determining the fundamental model atmosphere parameters (T eff, logg, v t,[fe/h]) of solar-type stars, which is based on the equivalent widths for a well-chosen set of Fe I and Fe II lines. The basic principle of this method is to find the solution in the (T eff,logg, v t ) space, which minimizes the sum of the dispersion of the Fe I abundances and the square of the Fe I Fe II abundance difference. An application of this code to the observed solar equivalent widths turned out to yield satisfactory results, which are more or less consistent with the actual parameters of the Sun. The numerical performance of this approach as well as the errors involved in the resulting solutions are discussed. Key words: methods: numerical stars: abundances stars: atmospheres stars: fundamental parameters stars: solar-type 1. Introduction In a model-atmosphere analysis to obtain information on stellar photospheric chemical abundances from spectral lines of various elements, one has to first determine the parameters characterizing the atmospheric model; i.e., T eff (the effective temperature), logg (the surface gravity), v t (the microturbulent velocity dispersion), and [X] (the metallicity which is usually represented by the abundance of Fe relative to the Sun; i.e. [Fe/H]). In principle, these parameters should desirably be established from the spectrum itself to be analyzed for each star by requiring that measurable quantities (e.g., strengths of spectral lines, wing profiles of strong lines, depth ratios of specific line pairs, etc.) calculated based on the model satisfactorily match the observations. Meanwhile, it appears that recent investigators in this field tend to prefer an easier way for evaluating these atmospheric parameters, rather than making an effort using such a steady and strict approach. This is because (a) well-calibrated T eff -determination methods from photometric colors are nowadays available, thanks to progress in energy distribution calculations from model atmospheres as well as in the evaluation method of T eff s for calibration stars (e.g., direct angular diameter measurements, IR flux method); (b) log g values can be reliably determined from absolute magnitudes (with estimated mass values) for many stars, for which Hipparcos parallaxes are known; and (c) even useful empirical formulas for v t are available (e.g., Nissen 1981; Edvardsson et al. 1993), which yield reasonable values. We should note, however, that those practical methods may not be regarded as being superior to the above-mentioned purely spectroscopic one. Namely, for our purpose of abundance determination, what we need is to choose the most appropriate atmospheric model (among the various ensembles specified by T eff,logg, andv t as useful indices) which has the (T,P) structure as close as to that of a real star and well reproduces its observed spectrum. From this point of view, the approach of using spectroscopically established T eff or log g values (even if they are not the true values!), which are based on the spectrum in question along with the adopted model atmospheres (to be used also in the abundance derivation), should be definitely favorable as compared to using the parameter values determined by other methods irrespective of the spectrum at hand, since the latter does not necessarily guarantee the consistency between theory and observation. (Note that we are not sure whether currently available atmospheric models represent the actual structures, especially in optically-thin line-forming regions.) In addition, there is a decisive merit in the spectroscopic method of parameter determinations: it does not suffer any effect of interstellar extinction/reddening unlike the cases of (a) or (b) mentioned above, and thus applicable to any stars no matter how they are distant from us. Nevertheless, there are some weak points in such a spectroscopic method of parameter determinations. First of all, it requires considerable efforts and is time-consuming. Since these atmospheric parameters are more or less dependent on each other (i.e., observable quantities are complicatedly affected by any of these in the strict sense), some of them should be simultaneously treated (e.g., T eff and log g determination by using the T g plane) and iterative procedures may usually be necessary. Also, during the course of this procedure, abundance calculations have to be repeated each time for slightly different parameter sets, and should be checked by plotting the results on the figure, which is quite laborious. Second, the determination more or less depends on one s subjective view, since it requires finding the intersection point of the lines/curves on the T g

2 452 Y. Takeda, M. Ohkubo, and K. Sadakane [Vol. 54, plane or an eye-inspection judgement whether any systematic tendency exists on the abundance diagram; thus, some uncertainties are inevitably involved with the estimated results in this sense. Considering this situation, we realized the necessity of developing an efficient method for spectroscopic determinations of model atmosphere parameters, by which the establishment of T eff,logg, v t,and[fe/h] can be done much more easily and quickly compared to the old approach, while being hardly affected by any subjective judgement. Motivated by this consideration, a program has been written for this purpose, which numerically determines these parameters as output results, for a given set of equivalent widths of well-chosen Fe I and Fe II lines as input data. Because of our present interest, targets of our program are limited to solartype stars of late-f through late-g spectral types with metallicities corresponding to the Galactic disk, though application to stars of other types may be easily done by replacing the precomputed data grid of the equivalent-widths by some other appropriate ones. In this article we report on the results of our first application of this method to the Sun, the most intensively studied star so far. Since its fundamental parameters are already well known to high precision, comparisons of the resulting solutions with such available information must be an important touchstone for our program. Further results of applications to other stars will be described in a separate paper. 2. Basic Principles for Parameter Determinations In traditional spectroscopic determinations of T eff,logg,and v t of solar-type stars, the observed equivalent widths of Fe I and Fe II lines are usually used while requiring three constraint conditions: (i) excitation equilibrium, (ii) ionization equilibrium, and (iii) matching the curve of growth shape, which are explained below T eff /logg-dependence of Fe Abundances First, we briefly describe how the resulting Fe abundances depend on a choice of T eff or logg in solar-type stars. In the atmospheres of solar-type stars in question, most Fe atoms are once-ionized and only a small fraction of them remain neutral. In such a case, the number population of neutral atoms at level i (n I i ) is scaled as n I i ɛt 3/2 n e e (χ ion χ i )/(kt ), (1) where ɛ is the Fe abundance, n e is the electron density, χ ion is the Fe ionization potential of 7.9 ev, χ i is the excitation potential of level i, andk is the Boltzmann constant. Meanwhile, that for level j of ionized atoms is written as n II j ɛe χ j /(kt ). (2) These relations, (1) and (2), can be derived from the Saha Boltzmann formula (see, e.g., subsubsection in Takeda 1991). Then, the dependence of the line-opacity (l) upon T eff and logg may be expressed as li I ɛt 3/2 eff g 1/3 e (χ ion χ i )/(kt eff ) (3) and lj II ɛe χ j /(kt eff ), (4) whereweputt T eff and used the approximation that the atmospheric density (pressure) roughly scales as g 1/3 (see, e.g., Gray 1992). Actually, the observed line-strength is determined by the ratio of the line opacity (l) to the continuum opacity (κ); and since the H opacity mainly contributing to the latter scales with n e similarly to li I (along with a weak T -dependence), the g-dependence for l I /κ tends to be cancelled out while a new g 1/3 -dependence appears for l II /κ. Consequently, the T eff -andg-dependence of the Fe abundance (ɛ) resulting from a given equivalent width is roughly written as ɛ I T +3/2 eff e (χ ion χ i )/(kt eff ) (5) and ɛ II e + χ j /(kt eff ) g +1/3. (6) In other words, the Fe abundance derived from Fe I lines increases with an increase in T eff, while it is rather insensitive to a change in log g. On the other hand, the abundance from Fe II lines decreases with an increase in T eff, while it increases as logg becomes higher Excitation Equilibrium According to equations (5), the sensitivity of ɛ I to a change in T eff is larger as the excitation potential (χ) becomes lower. By utilizing this fact, we can give a constraint on T eff by requiring that the abundances derived from Fe I lines, for which the values of the excitation potential spread over a sufficiently wide range, should be independent on χ. Of course, such a requirement of χ-independence may in principle be applied to Fe II lines (i.e., higher-excitation lines are more T eff -sensitive in this case). However, this is not very practical in the case of solar-type stars, where the number of available Fe II lines is not sufficiently large (i.e., the range of χ is not wide enough) Ionization Equilibrium Equations (5) and (6) indicate that the T eff -sensitivity for Fe I and Fe II is just the opposite to each other; i.e., ɛ I / T eff > 0 and ɛ II / T eff < 0. On the other hand, regarding the response to a variation in log g, we can state from the results in subsection 2.1 that ɛ I / g 0, while ɛ II / g > 0. Accordingly, the requirement, that the averaged abundance derived from Fe I lines and that from Fe II lines should be equal, can impose a strong constraint on both T eff and logg Adjustment of the Microturbulence According to its definition, the microturbulent velocity (v t ) is the quantity incorporated in the Doppler width of the lineopacity profile, which was introduced to increase the strength of the lines near or on the flat-part of the curve of growth, so as to bring the calculated strengths of saturated lines into consistency with the observed ones. Therefore only the abundances derived from more or less saturated stronger lines are affected by the choice of this parameter, while those from sufficiently weak lines on the linear part of the curve of growth are essentially v t -independent. Hence, as usually done, this parameter

3 No. 3] Spectroscopic Determination of Stellar Atmospheric Parameters 453 can be determined by requiring that the abundances from Fe I lines do not show any dependence on their equivalent widths. Though Fe II lines may in principle be used for this purpose, they are not suitable for solar-type stars for the same reason as mentioned in subsection 2.2 (i.e., paucity of the available lines) Conditions to Be Satisfied Based on what has been mentioned above, we summarize the three requirements as follows, by which the three atmospheric parameters (T eff,logg,v t ) should be established from the equivalent widths of the selected Fe I and Fe II lines: (1) The abundances derived from Fe I lines should not show any dependence on the lower excitation potential (χ). (2) The averaged abundance over those derived from Fe I lines and that from Fe II lines should be equal. (3) The abundances derived from Fe I lines should not show any dependence on the equivalent widths (W ). 3. Computational Program For the purpose of obtaining the solutions of (T eff,logg, v t ) numerically which satisfy the conditions (1) (3) simultaneously, for given data of the observed equivalent widths of Fe I and Fe II lines, we have developed a computer program, named TGV, 1 which is intended to be applied mainly to solar-type stars. In this section we describe the details of this program. Hereinafter, we abbreviate the logarithmic Fe abundance, logɛ Fe by A (in the usual normalization of logɛ H =12.00), with a subscript 1 or 2 to indicate that the abundance was derived from Fe I or Fe II, and the mean abundance averaged over lines as well as the r.m.s. standard deviation around the mean is denoted by A and σ, respectively Formulation of the Problem In order to search for the solution which fulfills constraints (1) (3) postulated in subsection 2.5, we formulated the problem in a slightly different way. That is, the two conditions, (1) and (3), are combined into one simple requirement, that the standard deviation, σ 1, around the mean A 1 should be minimized. Meanwhile, requirement (2) may be simply written as A 1 = A 2. Accordingly, we set up the problem to be solved as follows. We first define the dispersion function D 2 of three arguments, D 2 (T eff,logg,v t )as D 2 σ1 2 +( A 1 A 2 ) 2. (7) Then, the solution that we seek for may eventually be obtained as the (Teff,logg,vt ), at which the D 2 function is minimized 1 The source codes (written in Fortran 77/90) as well as the executable program files running on Windows are available from the anonymous FTP site of ftp:// (IP address: ; subdirectory: /Users/takeda/tgv/) for those who have interest in using it. 2 This function may be defined more generally by introducing arbitrary coefficients (c 1, c 2, c 3 )asd 2 (σ c 1σ 2 2 )+c 2( A 1 A 2 + c 3 ) 2,where c 1 and c 2 are the weight coefficients, and c 3 is a correction introduced to allow for the non-lte ionization equilibrium (which might become significant in metal-poor stars). However, we assume c 1 =0,c 2 =1,andc 3 =0 throughout this paper. in the 3D space of these three variables Selection of Line Data How should we choose the Fe I and Fe II lines to be used is very important for this approach of (T eff, log g,v t ) determination, since any improperly selected set may lead to incorrect solutions or may even end up with numerical instabilities. Namely, all lines should have sufficiently reliable gf values, and should be hardly affected by any blending so as to assure precise measurements of equivalent widths. We, therefore, decided to invoke the line set published by Grevesse and Sauval (1999), who used carefully examined 65 Fe I lines and 13 Fe II lines of high quality with the most reliable experimental gf values to date for their determination of the solar Fe abundance, which they derived to be 7.50 ± Regarding the relevant line data (wavelengths, lower excitation potentials, and gf values), the values taken from their table 1 were used unchanged, which are also listed in table 1 of this paper Data Calculation In order to find the solution numerically, it is necessary to evaluate the D 2 function defined by equation (7) quickly for any given combination of (T eff,logg,v t ). For this purpose, we first computed the logarithmic equivalent width (logw )forvarious combinations of these three parameters as well as for a sufficiently wide range of the Fe abundance. Regarding the model atmospheres, we exclusively used the ATLAS9 models computed by Kurucz (1993a). Also, his WIDTH9 program (though modified in various respects by Y. T.) was used for calculations of the equivalent widths of Fe I and Fe II lines, with the line data given in table 1. We applied an enhancement factor of 2 to the classical value (adopted as the default in WIDTH9; cf. Leushin, Topil skaya 1987; Unsöld 1955) of the van der Waals damping width (Γ 6 ) for all lines, according to the results of Anstee et al. (1997). For the radiation damping width (Γ rad ) and the quadratic Stark effect damping width (Γ 4 ), we applied the default treatments of the WIDTH9 program (cf. Leushin, Topil skaya 1987), though they are of minor importance in solar-type stars. Practically, for a given metallicity, we selected 81 models which are combinations of 9 T eff values (from 4750 K to 6750 K with a step of 250 K), 9logg values (from 1.0 to 5.0 with a step of 0.5). Then, for each of the 81 models, the logw values of all 78 lines were computed for any combinations of 12 v t values (from 0.0kms 1 to 5.5kms 1 with a step of 0.5kms 1 )and 18 A values (from 6.4 to 8.1 with a step of 0.1). The results were written out in a data file. At present, five such data files of logw are available corresponding to different model metallicities: 0.0 (solar metallicity) models for flux logw,0.0(solar metallicity) models for disk-center logw, +0.5dex models for flux logw, 0.5dex models for flux logw,and 1.0dexmod- els for flux logw Function Minimization Now that the grid of logw values has been calculated and is available in a file, our program first reads these data into memory, so that it may quickly evaluate logw of a line for any combinations of (T eff,logg, v t, A) by interpolation. Alternatively,

4 454 Y. Takeda, M. Ohkubo, and K. Sadakane [Vol. 54, Table 1. Data of the solar equivalent widths and the resulting Fe abundances. No. λ χ loggf W flux T A W flux M A W d.c. G A T + g + v + (Å) (ev) (må) (må) (må) [Fe I lines]

5 No. 3] Spectroscopic Determination of Stellar Atmospheric Parameters 455 Table 1. (Continued) No. λ χ loggf WT flux A WM flux A WG d.c. A T + g + v + (Å) (ev) (må) (må) (må) [Fe II lines] Note. The 1st column presents the assigned line number. The data of the spectral lines (wavelength, lower excitation potential, oscillator strength) given in columns 2 4 were exclusively taken from Grevesse and Sauval (1999). The adopted equivalent widths and the resulting Fe abundances (in the usual normalization of H = 12.00) corresponding to the final atmospheric parameters (cf. table 2) are shown in columns 5 10 for each of the three different data sets: the data from our own measurements on Kurucz et al. s (1984) KPNO solar flux spectra (WT flux ), those of Meylan et al. s (1993) based on the same KPNO flux spectra (WM flux ), and those used by Grevesse and Sauval (1999) based on the disk-center spectra of Delbouille et al. s (1973) Jungfraujoch atlas (WG d.c. G d.c. eff =+25K, log g =+0.08, and v t =+0.13 km s 1 (i.e., statistical uncertainties of these parameters, cf. table 2) are also given in columns Note that the equivalent width (4.9 må) of the Fe I line at Å given by Meylan et al. (1993) was not used, which may presumably be erroneous. for any combination of (T eff,logg,v t,logw ), the corresponding A (abundance) can be inversely determined. Therefore, since log W values are already given as input data, we can compute A and σ, and thus the function D 2 from equation (7), for any point in the (T eff,logg,v t ) space. Regarding the numerical algorithm for finding the solution (Teff,logg,vt ) which minimizes the function D 2, we adopted the downhill simplex method (subroutine AMOEBA) described in Press et al. (1992). For an arbitrary set of initial input values of (T eff, log g,v t ), this method searches for the desired D 2 -minimizing solution by iteration, where the judgement of minimum is done according to a specified tolerance parameter (FTOL). Since we found that convergence was occasionally not attained (i.e., endless loop) for an FTOL value as small as 10 6, we somewhat mitigated this criterion and used FTOL = 10 4, by which a reasonably quick convergence and numerical stability was accomplished at any time. As mentioned above, for a given initial solution set of (T 0 eff, log g0,v 0 t ), the program TGV is designed to compute the D 2 -minimizing solution while varying simultaneously all these three parameters as independent variables. In addition, however, we incorporated in this program two-variables mode (while the other one being fixed and unchanged from the initial solution) and one-variable mode (the other two being fixed unchanged), which actually turned out to be of practical use (cf. the next subsection) Iteration Procedure After some test calculations, we found that the threevariables mode (allowing all three T eff,logg, andv t to vary simultaneously) alone turned out not to be very successful; that is, it occasionally stabilized at an undesirable solution, especially with respect to v t. We, therefore, realized that it is not always practicable to arrive directly at the goal by using the three-variables mode only once, especially when starting from a solution widely discrepant from the true one; we thus chose to proceed in an alternative (iterative) way instead, as explained below.

6 456 Y. Takeda, M. Ohkubo, and K. Sadakane [Vol. 54, According to what we described in section 2, the coupling of T eff and log g is so strong that they had better be simultaneously treated, while the effect of v t is only W -dependent and not directly related to T eff or logg. Hence, we decided to handle (T eff,logg)andv t separately, while applying the two-variables mode and one-variable mode interchangeably. Suppose that we have solutions resulting from the previous iteration i 1, (T i 1 eff,logg i 1,andvt i 1 ). These are used as starting solutions for the current iteration i, which consists of two steps: (A) We first apply the two-variables mode to T eff and logg, which yields revised solutions of these two parameters. At this stage, the current solutions are (Teff i,loggi,andvt i 1 ), which are regarded as being the starting solutions for the next step (B). (B) Then, only v t is varied in the one-variable mode while the other two are fixed, which yields a revised solution vt i for this iteration i. As such, this iteration i is completed with the updated solution (Teff i,loggi,andvt i ), which may be used for the next iteration. It was found that repeating these two consecutive steps satisfactorily yields the correct solution fairly quickly, as described in subsection 4.2 for the case of the Sun. 4. Application to the Sun In order to examine the performance of the numerical algorithm and to check the correctness of the resulting solutions, we first applied our program to the Sun, for which very highquality spectra are available and its fundamental atmospheric parameters are well known Observational Data Regarding the observational equivalent-widths data of the Sun, we used three different kinds of data sets for comparison purposes. The first is our own measurement on the Solar Flux Spectrum Atlas published by Kurucz et al. (1984) [or Kurucz (1993b) for the electronic version], which we hereinafter abbreviate as WT flux. While inspecting the target 65 Fe I and 13 Fe II lines taken from Grevesse and Sauval s (1999) line list (cf. subsection 3.2), we tried to measure the equivalent widths of as many lines as possible, as far as reliable measurements are possible. As a result, we could obtain the equivalent widths for 59 Fe I and 13 Fe II lines. We applied the Gaussian fitting technique for evaluations of WT flux, except for two Fe I lines at Å and Å, for which the direct integration method was adopted. The placement of the continuum level was determined by our own eye-judgement inspecting a 10Å interval of the spectral data around the line to be measured (i.e., we did not necessarily adopt the unit-intensity level of the atlas data). The second is that of Meylan et al. (1993), who also measured the equivalent widths for a number of well-behaved lines by fitting the Voigt function on the same Solar Flux Spectrum Atlas (Kurucz et al. 1984) as used by us. Among their published data, 30 Fe I lines and 7 Fe II lines are common to those 78 lines of Grevesse and Sauval (1999), which we can use (hereinafter called WM flux flux ). The comparison of our WT and their WM flux is shown in figure 1. We can see from this figure Fig. 1. Comparison of the solar equivalent widths measured by us with those given by Meylan et al. (1993), both being based on the same observational data of the Solar Flux Spectrum Atlas (Kurucz et al. 1984). Filled circles Fe I lines, open circles Fe II lines. that our values are systematically smaller than theirs, which we suspect to be attributed to the difference in the adopted continuum position as well as to the Voigt-fitting technique which they adopted. We discuss this point again in the next subsection. The third is naturally the equivalent-width data of 65 Fe I and 13 Fe II lines adopted by Grevesse and Sauval (1999), which we call WG d.c.. Note, however, that their measurement was made on the Jungfraujoch atlas of the disk-center solar spectrum (Delbouille et al. 1973), unlike WT flux or WM flux. All of these equivalent-width data (WT flux, W M flux, W G d.c. )used in this study are given in table Results Our program was applied to those three sets of the Fe I and Fe II equivalent widths mentioned above to determine (T eff, log g, andv t ) for each case. Regarding the theoretically computed logw grid to be read in the program (cf. subsection 3.3), we naturally used the data corresponding to the flux calculation on 0.0 (solar metallicity) models for WT flux and WM flux,and those for the disk-center calculation on 0.0 (solar metallicity) models for WG d.c.. Starting from the trial (initial) solutions of (5000 K, 3.0, 3.0kms 1 ), which were intentionally chosen as largely discrepant values from the true ones, we repeated the iteration (consisting of two steps as explained in subsection 3.5) ten times. This sufficed to accomplish an almost complete convergence of the solutions (changes at the final iteration are 1K, 0.01, 0.01kms 1 ). In order to demonstrate this, the convergence performances of additional test calculations for the WT flux case are graphically depicted in figure 2, where we can see how the solutions converge to essentially the same values

7 No. 3] Spectroscopic Determination of Stellar Atmospheric Parameters 457 Table 2. Solutions of the atmospheric parameters of the Sun. Data source Type T eff logg v t N 1 A 1 σ 1 e 1 N 2 A 2 σ 2 e 2 (K) (cms 2 ) (kms 1 ) Our own flux measurement ±25.0 ±0.080 ±0.13 ±0.031 ±0.037 Meylan flux et al. ±30.0 ±0.075 ±0.17 ±0.041 ±0.034 Grevesse disk & Sauval center ±25.0 ±0.080 ±0.13 ±0.028 ±0.033 Note. These solutions correspond to the three kinds of different data sets given in table 1 (WT flux, M flux,andwd.c. G ). N, A, σ,ande denote the number of the lines, the averaged Fe abundance, the standard (r.m.s.) deviation, and the probable error ( σ/ N), respectively, where the suffix 1 is for the Fe I lines and 2 is for the Fe II lines. The values following the ± sign denote the estimated statistical errors in T eff,logg, andv t (cf. subsection 5.2 for their definitions) and the corresponding uncertainties in A 1 and A 2. Fig. 2. Graphical display of how the solutions are converged during the run of the iterative procedures described in subsection 3.5. The results for the four test cases are shown, each starting with different initial solutions of (T eff,logg,v t ): circles (5000 K, 3.0, 3.0kms 1 ), triangles (5000 K, 5.0, 0.0kms 1 ), squares (6500 K, 3.0, 3.0kms 1 ), and inverse triangles (6500 K, 5.0, 0.0kms 1 ). starting from four different initial solutions. The converged solutions of (T eff,logg, andv t ), along with the corresponding A (averaged Fe abundance), σ (standard deviation), and e ( σ/ N; probable error) for Fe I and Fe II, aregivenin table 2. The A vs. χ and A vs. W plots (corresponding to these converged values of atmospheric parameters) are also displayed in figure 3, where we can recognize that the three requirements, (1) (3), postulated in subsection 2.5 are reasonably fulfilled. Comparing the resulting T eff and logg values in table 2, with the true solar parameters (5773 K, 4.44), we may state that the consistency is to a satisfactory level for the cases of WT flux and WG d.c., considering the purely spectroscopic nature of this method. However, we note that the results from Meylan et al. s (1993) data (WM flux ) are appreciably different from the other two cases, and the deviations from the true values can not be neglected in the sense that these data yield too high T eff and log g values. This tendency is also apparent when we inspect the A values: namely, while reasonable values of 7.5 (cf. Grevesse, Sauval 1999) were obtained for WT flux and WG d.c., an apparently larger solution of 7.7 was derived from the WM flux data. This difference may be attributed to the systematic difference between WT flux and WM flux (cf. figure 1). Although we can not confidently decide which is more reasonable, we suspect that Meylan et al. s (1993) data (WM flux ) may be somewhat overestimated (presumably due to their adopted continuum level). Actually, Takeda (1995) already noticed this discrepancy in his (continuum-independent) line-profile analysis of solar spectral lines, suggesting that the continuum level adopted in Kurucz et al. s (1984) Solar Flux Atlas may be somewhat higher by 0.5% for weaker lines and 2% for stronger lines, which leads to an abundance overestimation amounting to dex (cf. figure 8 therein). In addition, their application of the Voigt function even to weak lines, which may not have been very adequate in spite of what they intended, may partly be responsible for this discrepancy. In any case, we feel that the results derived from the WM flux data had better be viewed with caution. 5. Error Estimation Then, how much uncertainties are involved in the derived solutions of T eff,logg,andv t? Although the problems of systematic errors (due to inadequate modeling, errors in the adopted atomic parameters, etc.) are not considered here, we discuss

8 458 Y. Takeda, M. Ohkubo, and K. Sadakane [Vol. 54, Fig. 3. Abundance vs. excitation potential relation (upper panels; a, b, c) and abundance vs. equivalent width relation (lower panels; a,b,c ), plotted from the results given in table 1, which correspond to the final solutions of the atmospheric parameters (cf. table 2) for the adopted three different dataset of equivalent widths. The position of the final (averaged) Fe abundance is indicated by the horizontal dashed line. Filled circles Fe I lines, open circles FeII lines. the uncertainties which inevitably stem from the statistical scatter (e.g., such that found in the A vs. χ or A vs. W plots; cf. figure 3), the effect caused by the use of different line set, and the metallicity effect Behavior around the Converged Solutions Figure 4 shows how the function D 2 [defined by equation (7)] as well as the related quantities [σ1 2, σ 2 2,( A 1 A 2 ) 2, A 1, A 2 ] vary when each of the T eff,logg,andv t values are changed around the converged solutions for the WT flux case. This figure contains wealthy information and provides us with an intuitive insight, regarding how the solutions are determined and to what extent of ambiguity is involved with the solution of each parameter Statistical Uncertainties We also tried to evaluate the errors in a more quantitative manner. The intrinsic uncertainty of the solution may be defined as the extent of the range, within which any change in the relevant parameter does not cause any substantial influence on the judgement that all of the three requirements (1) (3) (subsection 2.5) are fulfilled. (In other words, the appearance of figure 3 is hardly changed no matter how the parameter is changed within this range.) Then, what should be defined is the critical conditions corresponding to the breakdown of each of these three requirements [(1), (2), and (3)], which we postulate as follows: First, we can fit the A 1 vs. χ relation as well as the A 1 vs. W relation by the linear-regression line, A 1 = a + bχ (8) and A 1 = p + qw, (9) where a, b, p, andq are the coefficients to be determined numerically. We then consider that the χ-independence of A 1 is fulfilled when (χ max χ min ) b <σ 1, (10) where χ max and χ min are the maximum and minimum values

9 No. 3] Spectroscopic Determination of Stellar Atmospheric Parameters 459 Fig. 4. Graphical display of the variations of the dispersion function defined by equation (7) along with its related components (upper panels; a, b, c), or of the mean Fe abundances ( A 1, A 2 ) derived from Fe I and Fe II lines (lower panels; a,b,c ), which result from perturbing each of the T eff,logg, and v t around the converged solution of (5719 K, 4.350, 0.86 km s 1 ) obtained from our WT flux data (cf. table 2). In the lower panels, the ranges of ±σ (standard deviation) and ±e (probable error) are also shown by the dotted and dashed lines around the mean abundances (thick solid lines), respectively. Note that in panel c the curve of A 2 is vertically offset by 0.2 dex,since A 1 and A 2 are almost indiscernible with each other in this case. of χ. Similarly, the W -independence of A 1 is satisfied when (W max W min ) q <σ 1, (11) where W max and W min are the maximum and minimum values of W. The equality for A 1 and A 2 is considered to still hold as long as A 1 A 2 <e 1 + e 2. (12) Now, when we increase T eff progressively from the converged solution value of Teff, at which conditions (10) (12) are naturally satisfied, we eventually come to a temperature point (T eff + ) where any one of these (10), (11), and (12) is not fulfilled. Also, the lower temperature limit (T eff ) can be defined in the same manner on the other side. Then we may evaluate the uncertainty of T eff as ±(T eff + T eff )/2. Similarly, the uncertainties in log g and v t can be estimated in the same way. Such evaluated ambiguities are given in table 2 (i.e., values with the ± sign), where the uncertainties in the A values corresponding to these errors in T eff,logg, andv t are also presented, which were computed as [( A T ) 2 +( A g ) 2 +( A v ) 2 ] 1/ Effect of the Adopted Line Set After some test calculations, we noticed that the results are rather critically influenced by which line set is to be used. As a matter of fact, it is not uncommon that even removing only one line may cause appreciable changes in the converged solutions. In order to investigate this problem, we performed the following numerical experiment. As shown in table 2, our WT flux data of 59 Fe I lines and 13 Fe II lines yielded the solutions of (5719 K, 4.35, 0.86kms 1, 7.48) for (T eff,logg, v t, A ). It is interesting to see what kind of solutions will be obtained if we use a sub-set of these lines with a reduced size, which is constructed by excluding some number of lines from this fiducial set of (N 1,N 2 )=(59, 13) (N 1 and N 2 are the numbers of Fe I lines and Fe II lines, respectively). Therefore, three kinds of sub-sets were considered, which comprise (42, 9), (18, 4), and (6, 1) lines, each corresponding to 70%, 30%, and 10% of the original size, respectively. By randomly selecting the lines included in each subset, we repeated the experiments (just in the same way as was done in subsection 4.2)

10 460 Y. Takeda, M. Ohkubo, and K. Sadakane [Vol. 54, Fig. 5. Distribution of the converged solutions for T eff,logg, v t,and A (= A 1 = A 2 ) for three kinds of tests based on the solar WT flux values measured by us, where the numbers of adopted Fe I and Fe II lines (N 1,N 2 ) are reduced to (42,9), (18,4), and (6,1), which means that the line dataset is downsized to 70%, 30%, and 10% of the original set of (59,13). Each panel shows 500 solutions resulting from the experiments with randomly chosen lines (from the original set), while keeping the prescribed number of lines (N 1,N 2 ). The solutions are plotted in order of the experiment number.

11 No. 3] Spectroscopic Determination of Stellar Atmospheric Parameters 461 until 500 solutions were obtained for each case. 3 The results are displayed in figure 5. We can evidently see from this figure that the scatter (diversity) of the solution grows as the number of the used lines becomes smaller: the r.m.s. deviation for T eff is 18, 45, and 246 K; that for logg is 0.06, 0.16, and 0.83; that for v t is 0.05, 0.11, and 0.36 km s 1 ;andthatfor A is 0.02, 0.04, and Accordingly, it can be said that we had better use as many reliable lines as possible, if we try to accomplish a high precision in the determination of these parameters. Also, in the case of a differential analysis between two stars, the parameters of both stars should be established by using exactly the same line set,whichmayhelptominimizetheerrorofthis type Metallicity Effect Though the problem concerning the choice of the model metallicity is not directly relevant to the present application to the Sun, it is important when we apply our program to other stars, since the metallicity affects the ionization equilibrium between Fe I and Fe II through the electron pressure of the atmosphere. For example, if we use the log W data corresponding to 0.5dex and +0.5 dex solar metallicities (instead of 0.0 dex) for the parameter determination based on WT flux,we obtain (5663 K, 4.49, 0.58 km s 1, 7.445) and (5746 K, 4.11, 0.94kms 1, 7.505), respectively, which are appreciably different from the standard solution of (5719 K, 4.35, 0.86 km s 1, 7.479) obtained from the 0.0 dex (i.e., solar metallicity) data (cf. table 1). Therefore, if one does not care about the proper choice of the model-metallicity, errors amounting to such extents are to be expected. Generally speaking, in an application to other stars, it is desirable to evaluate the parameters twice by using two data files of log W corresponding to two different encompassing metallicities ([X] 1 and [X] 2 ; expressed in dex relative to the Sun), which yield two sets of solutions, (T eff,1,log g 1,v t,1, A 1 )and (T eff,2,logg 2,v t,2, A 2 ). Since the A value at the intermediate metallicity [X] is written as A = ( ) ( )/( ) A 2 A 1 [X] [X]1 [X]2 [X] 1 + A 1, (13) while A and [X] are related to each other by the relation A =[X] +7.50, (14) we can derive the final values of [X] f and A f by solving 3 When the variations of the parameters are not within < 10K, < 0.01, and < 0.01 km s 1 even after the 10th iteration, we discarded this solution while regarding it as being poorly convergent. Accordingly, the number of actually performed experiments is larger than 500 for each case (651, 599, and 950 for the 70%, 30%, and 10% cases, respectively). equations (13) and (14) as [X] f = [ 7.50 ( ) ( )] [X] 2 [X] 1 + [X]1 A 2 [X] 2 A 1 /[( ) ( )] A 2 A 1 [X]2 [X] 1 (15) and A f = [ 7.50 ( ) ( )] [X] 2 [X] 1 + [X]1 A 2 [X] 2 A 1 /[( ) ( )] A 2 A 1 [X]2 [X] (16) Then, the final solutions of (T eff,f, log g f,v t,f ) at the thusestablished metallicity ([X] f ) may be evaluated by interpolating the two solutions, (T eff,1,logg 1,v t,1 )and(t eff,2,logg 2,v t,2 ), just in a similar way to equation (13). 6. Conclusion We developed a numerical technique which quickly determines the fundamental atmospheric parameters of solar-type stars (T eff,logg, v t,and[fe/h]) based on purely spectroscopic observational data alone, in the sense that only the equivalentwidths of well chosen Fe I and Fe II lines are used. The basic constraints imposed for establishing these parameters are: excitation equilibrium of Fe I (i.e., the abundances from Fe I lines do not show any dependence upon the excitation potential), ionization equilibrium between Fe I and Fe II (i.e., the mean abundance derived from Fe I lines and that from Fe II lines are almost the same), and correct reproduction of the shape of the curve of growth (i.e., the abundances from Fe I lines do not show any dependence upon the equivalent width). Our program numerically finds the solution satisfying these three conditions by way of an optimization problem where the dispersion function, which is defined as the sum of the dispersion of the Fe I abundances and the square of the Fe I Fe II mean abundance difference, should be minimized. We applied our code to the observed solar equivalent widths of well-examined Fe I and Fe II lines taken from Grevesse and Sauval s (1999) line list, and found that the results are in reasonable agreement with the known atmospheric parameters for the Sun. The uncertainties involved in the derived solutions were also discussed. The application of this method to other solar-type stars will be described in a forthcoming paper, where the results will be compared with those derived from other techniques (e.g., colors, Hipparcos parallaxes, etc.). References Anstee, S. D., O Mara, B. J., & Ross, J. E. 1997, MNRAS, 284, 202 Delbouille, L., Roland, G., & Neven, L. 1973, Photometric Atlas of the Solar Spectrum from 3000 Å to Å(Liège: Institut d Astrophysique, Université deliège) Edvardsson, B., Andersen, J., Gustafsson, B., Lambert, D. L., Nissen, P. E., & Tomkin, J. 1993, A&A, 275, 101 Gray, D. F. 1992, The Observation and Analysis of Stellar Photospheres, 2nd ed. (Cambridge: Cambridge University Press), ch. 9 Grevesse, N., & Sauval, A. J. 1999, A&A, 347, 348 Kurucz, R. L. 1993a, Kurucz CD-ROM, No. 13 (Harvard-Smithsonian Center for Astrophysics)

12 462 Y. Takeda, M. Ohkubo, and K. Sadakane Kurucz, R. L. 1993b, Kurucz CD-ROM, No. 18 (Harvard- Smithsonian Center for Astrophysics) Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, Solar Flux Atlas from 296 to 1300 nm (Sunspot, New Mexico: National Solar Observatory) Leushin, V. V., & Topil skaya, G. P. 1987, Astrophysics, 25, 415 Meylan, T., Furenlid, I., Wiggs, M. S., & Kurucz, R. L. 1993, ApJS, 85, 163 Nissen, P. E. 1981, A&A, 97, 145 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Second Edition (Cambridge: Cambridge University Press), 402 Takeda, Y. 1991, A&A, 245, 182 Takeda, Y. 1995, PASJ, 47, 337 Unsöld, A. 1955, Physik der Sternatmosphären, 2nd ed. (Berlin: Springer), 333