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Astron. Astrophys. 334, 221-238 (1998)

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3. Analysis

3.1. Abundance analysis

Our method of abundance analysis closely follows that used in Paper I and by Gonzalez & Lambert in their study of several stars in the [FORMULA] Per cluster. Like those studies, this analysis is a differential one with respect to the Sun. Since the stars on our program are similar to the Sun in their physical characteristics, a differential abundance analysis with the Sun as the source of the gf -values (as opposed to laboratory gf -values) avoids many possible systematic errors. These include uncertainties in the treatment of the structure of the atmospheres, such as differences in the mixing length to scale height ratio or convection and chromospheric activity, especially starspots. 2 Following Paper I and Gonzalez & Lambert we employ the Kurucz (1993) model atmospheres; the effective temperature, [FORMULA], surface gravity, g, and depth-independent microturbulence parameter, [FORMULA], are estimated in the standard way using the Fe I and Fe II lines (details are discussed in Sect. 3.1.1).

The abundance analyses of the Group 1 stars are based on measurements of the moderate resolution spectra obtained with the 2.1 m telescope, while the analysis of [FORMULA] CrB is based on the moderate resolution spectrum obtained with the 2.7 m. The spectral lines used in the present analysis were selected from the linelist in Tables 12 and 13 of Gonzalez & Lambert and Table 1 of Paper I. Additional lines were added in the analysis of the Group 1 stars (Table 1) and their gf -values estimated using the equivalent widths ([FORMULA] 's) measured on the Solar Flux Atlas (Kurucz et al. 1984); additional lines in the analysis of [FORMULA] CrB (Table 2) are based on measurements of a spectrum of Vesta described in Paper I. Although the wavelength coverage is quite extensive and the quality of the spectra is very high, we have been very restrictive in our selection of lines for use in the abundance analysis. To be suitable, a spectral line must be unblended (except for the few rare instances where the line is critical to estimating the abundance of a given element) and have a symmetric profile in both the target star spectrum and the solar spectrum. The sample is restricted to mostly moderate strength lines with [FORMULA] 's between about 10 and 150 mÅ (weaker lines have a larger relative error in the [FORMULA] measurements due to noise, and stronger lines are more sensitive to errors in [FORMULA]). Comparing measurements of the same lines on spectra taken on different nights, we estimate that the average uncertainty in an [FORMULA] measurement to be about 1-2 mÅ for lines with [FORMULA] near 50 mÅ and about 2-3 mÅ for stronger ones. Such low errors were achieved by smoothing isolated lines (effectively increasing the S/N ratio), measuring some lines on the overlap regions of adjacent orders, and averaging [FORMULA] measurements from spectra obtained on different observing runs.


Table 1. Atomic data for lines used in the analysis
of the Group 1 stars and not listed by Gonzalez &
Lambert (1996)


Table 2. Atomic data for lines used in the
analysis of [FORMULA] CrB and not listed in Paper I

3.1.1. Model atmosphere selection and Fe abundances

While hundreds of Fe lines are present in the spectral regions observed, we selected only 30 Fe I and 5 Fe II for use in the analysis (the total number varies from star to star). The values of [FORMULA], [FORMULA], [FORMULA], and [Fe/H] have been estimated for each star (Table 3) with this sample of Fe lines using an updated version of the LTE line abundance code, MOOG (Sneden 1973). The atmospheric parameters are determined with an iterative procedure. We begin with an initial set of parameters equal to those determined in previous studies. All four parameters are adjusted in a systematic manner until the correlation coefficients between [FORMULA] (Fe I) and [FORMULA] and between [FORMULA] (Fe I) and [FORMULA] are zero and also [FORMULA] (Fe I) = [FORMULA] (Fe II). The range of the measured Fe I line strengths is sufficient to estimate [FORMULA] accurately. We note that the values of [FORMULA] we derived for all the stars but [FORMULA] Cnc and [FORMULA] CrB are the same as the solar value, as expected. The range of the lower excitation potentials, [FORMULA], is 1.0 to 5.0 eV for the Fe I lines, which is sufficient to estimate [FORMULA] accurately for each of the program stars. The typical uncertainties of our estimates of [FORMULA], [FORMULA], and [FORMULA] are 75 K, 0.1, and 0.1 km s-1, respectively (individual values are listed in Table 3). They lead to a typical uncertainty in [Fe/H] 3 of 0.06 dex, which was calculated with the sensitivities of the abundances to changes in atmospheric parameters (Table 5). The sensitivities of the [FORMULA] (Fe) versus [FORMULA] and [FORMULA] (Fe) versus [FORMULA] relationships to changes in [FORMULA] and [FORMULA], respectively, are shown in Fig. 1 for 51 Peg. The measured values of [FORMULA] for each line are listed in Table 4.


Table 3. Atmospheric parameters derived from the Fe-line analyses


Table 4. Equivalent widths of the program stars


Table 4. (continued) Equivalent widths of the program stars


Table 5. Sensitivities of calculated abundances to changes in model
atmosphere parameters for 51 Peg and [FORMULA] Cnc

[FIGURE] Fig. 1. The Fe I (filled circles) and Fe II (plus signs) abundances calculated using the model parameters given in Table 3 are plotted versus [FORMULA] and [FORMULA]. The solid lines are least-squares fits through the Fe I data points. The dotted lines in the first diagram are the least-squares fits when [FORMULA] is changed by [FORMULA] 250 K. The dotted lines in the second diagram are the least-squres fits when [FORMULA] is changed by [FORMULA] 0.2 km s-1.

As a check on the estimated uncertainties in the atmospheric parameters, we note that the removal of three high-weight Fe I lines (at 6574 Å, 6581 Å, and 6591 Å; all have small values of [FORMULA] and two have low values of [FORMULA]) from the analysis resulted in revisions to the atmospheric parameters within the range of the uncertainties. The only exception is [FORMULA] Cnc, for which the change in [FORMULA] was significantly larger than the estimated uncertainty in this parameter; this was due to the small number of Fe I lines (especially weak lines) measured in its spectrum. Overall, though, this shows that our solutions are robust and consistent with the quoted uncertainties. Finally, we should note that although our method of analysis closely follows that of Gonzalez & Lambert, our results are more precise than theirs due to our use of Fe I lines with a larger spread in [FORMULA] and [FORMULA]. Also, our results are more precise than those of Paper I due to the lower rotation velocities of our sample.

3.1.2. Other elements

Since the resonance line of lithium is blended with other lines in spectra of solar-type stars, we must employ spectrum synthesis methods to estimate its abundance. We used the linelist of Cunha et al. (1995, Table 7), modified slightly to reproduce the solar spectrum using the Kurucz solar model atmosphere; the spectral region synthesized spans 6700 to 6711 Å. The same stellar atmospheric parameters derived from the Fe-line analysis were used in producing the synthetic spectra. The line broadening was approximated with a Gaussian function with a width chosen so that the two strongest Fe I lines in the observed spectra are reproduced accurately. The lithium line is discernible by eye on all spectra, except those of [FORMULA] Cnc and 16 Cyg B, where it is not detectable at the level of the noise. Sample spectra and syntheses of the Li I region are shown in Fig. 2a and b for 16 Cyg A and B, for which we derive very different lithium abundances.

[FIGURE] Fig. 2. Portions of the spectra of 16 Cyg A and B containing the Li I line at 6707.8 Å (panel a). The strength of the Li I line differs significantly on the two spectra. Each spectrum is the average of three spectra obtained on different nights. The resultant S/N ratios are near 350. Shown in panel b is a comparison of a spectrum of 16 Cyg A with three syntheses, each differing only in the lithium abundance assumed.

The abundances of 15 additional elements were determined with the atmospheric parameters listed in Table 3. The individual line measurements are listed in Table 4, and the final abundances are given in Tables 6, 7, 8. As in the Fe abundance analyses, the uncertainties in the [X/H] values were calculated with the estimated uncertainties in the atmospheric parameters (Table 3) and the data in Table 5.


Table 6. Final adopted abundances for [FORMULA] Cnc, 51 Peg, 47 UMa, 70 Vir, and HD 114762


Table 7. Final adopted abundances for 16 Cyg A and B


Table 8. Final adopted abundances for [FORMULA] CrB

3.2. v sin i

To estimate the masses of the claimed planetary mass companions, we require, among other quantities, estimates of their orbital inclinations. Assuming that the orbital axis of a planet is aligned with the rotation axis of its parent star, an estimate of the projected stellar rotational velocity ([FORMULA]) can be used to constrain the orbital inclination. In the following we describe the derivation of [FORMULA] for [FORMULA] Cnc, 51 Peg, 47 UMa, 70 Vir, HD 114762 from high-resolution spectra (obtained with the 2.7 m telescope) and for [FORMULA] CrB from a moderate-resolution spectrum. In addition to the program star spectra, a spectrum of the sky was acquired to calibrate our technique with the known solar parameters.

We employ the Fourier transform method, supplemented by line profile synthesis, to estimate [FORMULA] from the stellar spectra. The instrumental, macroturbulent ([FORMULA]), microturbulent, rotational, and thermal broadening mechanisms are included in the analysis of a given line profile. The syntheses have been carried out with MOOG, which includes all the line broadening mechanisms; as in the abundance analysis, we make use of the Kurucz (1993) LTE model atmospheres. We have adopted the radial-tangential description of macroturbulent line broadening (Gray 1992), and we have approximated the instrumental broadening by a Gaussian function, its width determined from the emission lines in the Th-Ar comparison spectrum obtained immediately following each stellar observation. This approximation is a very close fit to the Th-Ar lines; however, even if it were not, it would not cause significant errors since the instrumental broadening is minor compared to the other line broadeners. The limb darkening coefficients, required for synthesizing the rotational profiles, have been interpolated from Fig. 17.6 of Gray (1992). The model atmosphere parameters used in the syntheses are the same as those estimated from the Fe-line analyses in Sect. 3.1.1, except for the Fe abundance and [FORMULA] (see below).

The Fe I lines at [FORMULA] 5379.586, 5638.249, 5731.762 Å were selected for analysis from Table 2 of Takeda (1995); these lines are unblended, moderate in strength, and have smaller than average and nearly identical [FORMULA] values. Our high-resolution spectrum of 51 Peg did not include these lines, since it was obtained with a different instrumental setup; we used the Fe I lines at [FORMULA] 6200.313, 6213.430, 6322.686 Å instead. All the lines selected for analysis appear symmetric upon visual inspection and do not have any other lines within about 0.3 Å of their line cores. The spectra were shifted to the rest frame and 0.6 Å sections containing the Fe I lines were isolated for further analysis. Each section contains the entire line profile along with a small amount of continuum.

The analysis method involves first producing a synthesis of a line and manually adjusting the values of [FORMULA], [FORMULA], [FORMULA] (Fe), and the continuum level until the residuals between it and the observed profile are minimized. Next, the Fourier amplitude spectrum of the observed profile is compared with the Fourier amplitude spectra of synthetic line profiles using a range of [FORMULA] and [FORMULA] values centered on the best-fit set. The synthetic profiles are generated by convolving the instrinsic stellar thermal profile with the instrumental, microturbulent, macroturbulent, and rotational broadening functions. We analyzed the solar spectrum first with [FORMULA] set equal to 1.7 km s-1 ([FORMULA] is the synodic rotation velocity of the Sun; Soderblom 1982) in order to estimate [FORMULA] and [FORMULA]. The value of [FORMULA] required to reproduce the solar Fe I line profiles is 0.4 km s-1. In their studies of the solar spectrum Gray (1977) and Takeda (1995) find that they must assume [FORMULA] = 0.5 km s-1 in order to reproduce the line profiles accurately. This is half the value we used in our abundance analyses of solar Fe I lines. The discrepancy is likely caused by the approximate nature of our line profile synthesis method and model-incompleteness (see Takeda et al. 1996 for a brief discussion of this problem). Regardless of its source, it is only a scientific problem, not a practical one. Both Soderblom (1982) and Hale (1995) find that [FORMULA] is only weakly dependent upon [FORMULA].

Treating both [FORMULA] and [FORMULA] as free parameters and fixing [FORMULA] at 0.4 km s-1, we obtain [FORMULA] and [FORMULA] km s-1 for the solar lines. 4 The final step in the analysis involves comparing the synthetic line profile with the observed profile again, this time using the parameters determined from the Fourier analysis. To obtain an adequate fit to observed solar line profiles, it was found necessary to increase [FORMULA] by 0.05 dex and [FORMULA] by 0.1 km s-1. We followed the same procedure in analyzing the line profiles of the other stars. We present the final results in Table 9 and sample Fourier amplitudes and synthetic spectra in Figs. 3 and 4, respectively. The quality of the spectra vary considerably, with those of the Sun, 70 Vir, and 47 UMa being the best and HD 114762 the poorest. The quality of the 51 Peg profiles is quite high, but the resolving power is only half that of the best spectra. The resolving power of the [FORMULA] Cnc spectrum is intermediate between that of 51 Peg and the other stars, but given that it is much cooler than the other stars and very metal-rich, the continuum is not nearly as smooth due to the presence of weak lines, which mimic noise. Zeeman broadening was not included in any of the analyses, since it has not been found to be a significant source of line broadening for stars hotter that G6 (Gray 1984). The low chromospheric activity of [FORMULA] Cnc implies that Zeeman broadening is probably not significant for this star, even though its spectral type is G8.


Table 9. Predicted [FORMULA] and measured [FORMULA] and [FORMULA] values

[FIGURE] Fig. 3. Fourier transforms of three Fe I lines in the high-resolution spectrum of 70 Vir. Also shown are the theoretical transforms calculated using different combinations of [FORMULA] and [FORMULA].
[FIGURE] Fig. 4. Synthetic profile (solid curve) of the Fe I line at 5379.59 Å in the spectrum of 70 Vir using [FORMULA] = 4.2 and [FORMULA] = 0.5 km s-1 ; the dashed curve corresponds to [FORMULA] = 4.2 and [FORMULA] = 1.5 km s-1.

We have calculated the expected value of [FORMULA] for each star based on the relation between it and the fundamental stellar parameters, as given in Gray (1992). Using Gray's (1992) Fig. 18.9 and the results of Gray (1984), we have generated an interpolation equation relating [FORMULA] to [FORMULA] and [FORMULA] and another relating [FORMULA] to [FORMULA] and [FORMULA], with residuals of [FORMULA] 0.2 and [FORMULA] 0.5 km s-1, respectively:


The zero points of these equations were set such that they yield the value of [FORMULA] that we estimated for the Sun (Table 9). The predicted values of [FORMULA] are consistent with the measured values, except for [FORMULA] Cnc, where the two predicted values are very different from each other.

We should note that the uncertainties we quote are formal. They are based on the scatter of the solutions for the individual lines. We have not taken into account possible systematic errors caused by velocity fields not included in our line modeling. The upper limits are also formal. The upper limit for 70 Vir is more secure than for [FORMULA] Cnc and HD 114762 due to the small scatter among the Fourier amplitude plots of the observed spectra. The extreme upper limit of [FORMULA] allowed by the data for 70 Vir is somewhat short of 1 km s-1, while for [FORMULA] Cnc and HD 114762 it might approach 2 km s-1. The close agreement between the predicted and measured values of [FORMULA] for all the stars but [FORMULA] Cnc is encouraging.

Due to the lower resolving power of our spectrum of [FORMULA] CrB, we did not attempt to determine both [FORMULA] and [FORMULA] for this star. Instead, we fixed [FORMULA] at 4.2 km s-1 from Eqs. 1 and 2. Using only the spectrum synthesis method with the 5379 and 5638 Å Fe I lines, we determined the best-fitting value of [FORMULA] to be about 1.5 km s-1.

Previous estimates of [FORMULA] exist for all our program stars. The quality of these estimates varies considerably depending on the quality of the spectra (due both to resolving power and S/N ratio), the number of lines analyzed, and the method used (line synthesis or Fourier transform); it seems that every work adopts a different approach in estimating [FORMULA]. We list in Table 10 the published [FORMULA] values. We also list in the table the mean adopted values giving all estimates equal weight and correcting Hatzes et al.'s (1997) and Francois et al.'s (1996) estimates by -0.2 km s-1. Hatzes et al. incorrectly used a solar [FORMULA] value of 2.0 km s-1 in calibrating their method, and Francois et al. incorrectly assumed the value of [FORMULA] in 51 Peg is the same as in the Sun.


Table 10. Published estimates of [FORMULA] and mean adopted values

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Online publication: May 12, 1998