V. V. Kovtyukh ^{1} - C. Soubiran ^{2} - S. I. Belik ^{1} - N. I. Gorlova ^{3}
1 - Astronomical Observatory of Odessa
National University and Isaac Newton Institute of Chile,
Shevchenko Park, 65014 Odessa, Ukraine
2 -
Observatoire de Bordeaux, CNRS UMR 5804, BP 89, 33270 Floirac, France
3 -
Steward Observatory, The University of Arizona,
Tucson, AZ 85721, USA
Received 6 May 2003 / Accepted 19 August 2003
Abstract
Line depth ratios measured on high resolution
(R=42 000), high S/N echelle spectra are used for the
determination of precise effective temperatures of 181 F, G, K Main
Sequence stars with about solar metallicity (
). A set of 105 relations is obtained which rely
on ratios of the strengths of lines with high and low
excitation potentials, calibrated against previously published
precise (one per cent) temperature estimates. The application
range of the calibrations is 4000-6150 K (F8V-K7V). The
internal error of a single calibration is less than 100 K, while
the combination of all calibrations for a spectrum of S/N=100reduces uncertainty to only 5-10 K, and for S/N=200 or higher
- to better than 5 K. The zero point of the temperature scale
is directly defined from reflection spectra of the Sun with an
uncertainty about 1 K. The application of this method to
investigation of the planet host stars properties is discussed.
Key words: stars: fundamental parameters - stars: planetary systems
The direct method to determine the effective temperature of a star relies on the measurement of its angular diameter and bolometric flux. In practice certain limitations restrict the use of this fundamental method to very few dwarfs. Other methods of temperature determination have errors of the order 50-150 K, which translates into the [Fe/H] error of 0.1 dex or larger. The only technique capable so far of increasing this precision by one order of magnitude, is the one employing ratios of lines with different excitation potentials . As is well known, the lines of low and high respond differently to the change in . Therefore, the ratio of their depths (or equivalent widths, EW) should be very sensitive temperature indicator. The big advantage of using line-depth ratios is the independence on the interstellar reddening, spectral resolution, rotational and microturbulence broadening.
The reader is referred to Gray (1989, 1994) and Gray & Johanson (1991) to learn more about the history and justification of the line ratio method. Applying this method to the Main-Sequence (MS) stars, they achieved precision as high as 10 K. The most recent works on the subject are by Caccin et al. (2002) who discuss the possible weak points of this technique for the case of dwarfs (see below), and the fundamental contribution by Strassmeier & Schordan (2000) who report 12 temperature calibrations for giants with an error of only 33 K.
So far however the line-ratio method has not been fully utilized for purposes other than just temperature estimation by itself. One of few applications is the chemical abundance analysis of supergiants, where it has proved the anticipated high efficiency and accuracy. Thus, Kovtyukh & Gorlova (2000, hereafter Paper I) using high-dispersion spectra, established 37 calibrations for the temperature determination in supergiants (a further study increased this number to 55 calibrations). Based on this technique, in a series of 3 papers, Andrievsky et al. (2002, and references therein) derived temperatures for 116 Cepheids (from about 260 spectra) at a wide range of galactocentric distances ( kpc) with a typical error 5-20 K. The high precision of this new method of temperature determination allowed them to uncover the fine structure in the Galactic abundance gradients for many elements. Even for the most distant and faint objects ( mag) the mean error in was no larger than 50-100 K, with maximum of 200 K for spectra with lowest S/N (=40-50).
Another example concerns T Tau stars. For young stars, uncertainties in reddening due to variable circumstellar extinction invalidate the photometric color method of effective temperature determination. Using 5 ratios of FeI and VI lines calibrated against 13 spectral standards, Padgett (1996) determined the effective temperature of 30 T Tau stars with a 1 uncertainty lower than 200 K.
The intent of this paper is to improve this technique, based on our experience of applying it to supergiants (Paper I and following publications), and expand it to the MS stars. The wide spectral range of ELODIE echelle spectra allowed to select many unblended lines of low and high excitation potentials thus improving the internal consistency of the method, whereas the large intersection between the ELODIE database and published catalogues of effective temperatures allowed to take care of systematic effects. We obtained a median precision of 6 K on derived for an individual star. The zero-point of the scale was directly adjusted to the Sun, based on 11 solar reflection spectra taken with ELODIE, leading to the uncertainty in the zero-point of about 1 K.
Temperature determined by the line ratio method may now be considered as one of the few fundamental stellar parameters that have been measured with an internal precision of better than 0.2%.
The investigated spectra are part of the library collected with the ELODIE spectrometer on the 1.93-m telescope at the Haute-Provence Observatory (Soubiran et al. 1998; Prugniel & Soubiran 2001). The spectral range is 4400-6800 Å and the resolution is R=42 000. The initial data reduction is described in Katz et al. (1998). All the spectra are parametrized in terms of , , [Fe/H], either collected from the literature or estimated with the automated procedure TGMET (Katz et al. 1998). This allowed us to select a sample of spectra of FGK dwarfs in the metallicity range . Accurate Hipparcos parallaxes are available for all of the stars of interest enabling to determine their absolute magnitudes M_{V} that range between 2.945 (HD 81809, G2V) and 8.228 (HD 201092, K7V). All the selected spectra have a signal to noise ratio greater than 100 (see Fig. 1). Further processing of spectra (continuum placement, measuring equivalent widths, etc.) was carried out by us using the DECH20 software (Galazutdinov 1992). Equivalent widths EWs and depths of lines were measured manually by means of a Gaussian fitting. The Gaussian height was then a measure of the line depth. This method produces line depths values that agree nicely with the parabola technique adopted in Gray (1994). We refer the reader to Gray (1994, and references therein) and Strassmeier & Schordan (2000) for a detailed analysis of error statistics.
Figure 1: Comparison spectra for two stars: solid line - a planet-host star HD 217014 (51 Peg), and dotted line - a non-planet star HD 5294. Within the limits of the errors, both stars have identical temperatures (5778 and 5779 K, respectively), but different metallicities. Spectral lines used in temperature calibrations are identified at the bottom with their wavelength, element, and lower excitation potentials in eV. We used ratios 6081.44/6089.57, 6085.27/6086.29, 6085.27/6155.14, 6089.57/6126.22, 6090.21/6091.92, 6090.21/ 6102.18, 6091.92/6111.65 and others. | |
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Following Caccin's et al. (2002) results, where a careful analysis of the anticipated problems for the Solar-type stars has been carried out, we did not use ion lines and high-ionization elements (like C, N, O) due to their strong sensitivity to gravity.
Gray (1994) showed that the ratio of lines VI 6251.82 and FeI 6252.55 depends strongly on metallicity. The reason is that the strong lines like FeI 6252.55 ( for the Sun) are already in the dumping regime, where the linearity of EW on abundance breaks down. In addition, as was shown in the careful numerical simulations by Stift & Strassmeier (1995), this ratio (of 6251.82 and 6252.55 lines) is also sensitive to rotational broadening. Significant effects were found for as small as 0-6 km s^{-1} (for solar-like stars). We therefore avoided using strong lines in our calibrations. Indeed, Gray (1994) concluded that, as expected, the weak-line ratios are free from the effects of metallicity. As to the effect of rotation, we should note that all objects in our sample are old Main Sequence stars with slow to negligible rotation ( km s^{-1}), which is comparable to the instrumental broadening.
Table 1: Program stars. Asterisks indicate stars with planets.
Thus, we initially selected about 600 pairs of 256 unblended SiI, TiI, VI, CrI, FeI, NiI lines with high and low excitation potentials within the wavelength interval 5300-6800 Å.
These lines have been selected according to the following criteria:
(1) the excitation potentials of the lines in a pair must differ as much as possible;
(2) the lines must be close in wavelength; it turned out though that calibrations based on widely spaced lines (including from different orders) show same small dispersion as the closely spaced lines. Therefore, we retained all pairs with a difference in wavelength up to 70 Å ( Å);
(3) the lines must be weak enough to eliminate a possible dependence on microturbulence, rotation and metallicity;
(4) the lines must be situated in the spectral regions free from telluric absorption.
The next step was to choose the initial temperatures for interpolation. This is a very important procedure since it affects the accuracy of the final temperature scale, namely, the run of the systematic error with (Fig. 3). There is an extended literature on MS stars temperatures. For 45 stars from our sample (see Table 1) we based the initial temperature estimates on the following 3 papers: Alonso et al. (1996, hereafter AAMR96), Blackwell & Lynas-Gray (1998, hereafter BLG98) and DiBenedetto (1998, hereafter DB98). In these works the temperatures have been determined for a large fraction of stars from our sample with a precision of about 1%. AAMR96 used the Infrared Flux Method (IRFM) to determine for 475 dwarfs and subdwarfs with a mean accuracy of about 1.5% (i.e., 75-90 K). BLG98 also have determined temperatures for 420 stars with spectral types between A0 and K3 by using IRFM and achieved an accuracy of 0.9%. DB98 derived for 537 dwarfs and giants by the empirical method of surface brightness and Johnson broadband (V-K) color, the accuracy claimed is 1%. Whenever 2 or 3 estimates were available for a given star, we averaged them with equal weights. These temperatures served as the initial approximations for our calibrations.
First, for the mentioned above 45 stars with previously accurately determined we plotted each line ratio against , and retained only those pairs of lines that showed unambiguous and tight correlation. We experimented with a total of nearly 600 line ratios but adopted only the 105 best - the ones showing the least scatter. These 105 calibrations consist of 92 lines, 45 with low ( eV) and 47 with high ( eV ) excitation potentials. Judging by the small scatter in our final calibrations (Fig. 2) and , the selected combinations are only weakly sensitive to effects like rotation, metallicity and microturbulence. This confidence is reinforced by the fact that the employed lines belong to a wide range of chemical elements, intensity and atomic parameters, therefore one can expect the mutual cancellation of opposite effects.
Each relationship was then fitted with a simple analytical function. Often calibrations show breaks which not can be adequately described even by a 5th-order polynomial function (see Fig. 2). Therefore, we employed other functions as well, like the Hoerl function ( , where , a,b,c - constants), modified Hoerl ( ), power low ( ), exponential ( ) and logarithmic ( ) functions. For each calibration we selected the function that produced the least square deviation. As a result, we managed to accurately approximate the observed relationships with a small set of analytic expressions. This first step allowed us to select 105 combinations, with an rms of the fit lower than 130 K, the median rms being 93 K. Using these initial rough calibrations, for each of the 181 target stars we derived a set of temperatures (70-100 values, depending on the number of line ratios used), averaged them with equal weights, and plotted these mean (with errors of only 10-20 K) versus line ratios again, thus determining the preliminary calibrations (for which the zero-point had yet to be adjusted).
We would like to point out that the precision of our calibrations varies with temperature. In particular, at high the lines with low become very weak causing the line depth measurement to be highly uncertain. Therefore, for each calibration we determined the optimum temperature range where the maximum accuracy is attained (no worse than 100 K), so that for a given star only a subset of calibrations can be applied.
What are the main sources of random errors in the line ratio
method? The measurement errors in line depths are mainly caused by
the continuum placement uncertainty and by the Gaussian
approximation of the line profile. In addition, the individual
properties of the stars, such as metallicity, spots, rotation,
convection, non-LTE effects and binarity may also be responsible
for the scatter observable in Fig. 2. The detailed analysis of
these and other effects can be found in Paper I, Strassmeier &
Schordan (2000) and in works by D. F. Gray. We estimate that
the typical error in the line depth measurement
is 0.02-0.05, implying an error in
temperature of about 20-50 K.
Figure 2: Our final calibrations of temperature versus line depth ratios r=R1/R2. The temperatures are shown as the average value derived from all calibrations available for a given star. The errors in temperature are less than the symbol size. The typical error in line ratio is 0.02-0.05. Position of the Sun is marked by the standard symbol. | |
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The mean random error of a single calibration is 60-70 K (40-45 K in the most and 90-95 K in the least accurate cases). The use of 70-100 calibrations reduces the uncertainty to 5-7 K (for spectra with S/N=100-150). Better quality spectra (R>100 000, S/N>400) should in principle allow an uncertainty of just 1-2 K. Clearly, time variation of the temperature for a given star should be readily detected by this method, since the main parameters that cause scatter due to star-to-star dissimilarities (gravity, rotation, [Fe/H], convection, non-LTE effects etc.) are fixed for a given star. The temperature variation of several degrees in mildly active stars may be produced by the surface features and rotational modulation, as for example has been documented for the G8 dwarf Bootis A (Toner & Gray 1988) and Dra (K0V, Gray et al. 1992).
The next stage is to define the zero point of our temperature scale. Fortunately, for dwarfs (unlike for supergiants) a well-calibrated standard exists, the Sun. Using our preliminary calibrations and 11 independent solar spectra from the ELODIE library (reflection spectra of the Moon and asteroids), we obtained a mean value of K for the Sun's temperature. Considering the Sun as a normal star, we adjusted our calibrations by adding 44 K to account for the offset between the canonical Solar temperature of 5777 K and our estimate. The possible reasons for this small discrepancy are discussed below.
Figure 3: Comparison between the temperatures derived in the present work and those derived by AAMR96 - squares, BLG98 - circles, and DB98 - triangles. The dashed line represents the linear fit to the data, and the solid line represents the one-to-one correlation. | |
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Table 1 contains our final determinations for 181 MS stars. Note that we added the 44 K correction to the initial calibrations in order to reproduce the standard 5777 K temperature of the Sun. For each star we report the mean , number of the calibrations used (N), and the standard error of the mean (). For comparison, we also provide as determined in Edvardsson et al. (1993, hereafter EDV93), AAMR96, BLG98 and DB98. Absolute magnitudes M_{V} have been computed from Hipparcos parallaxes and V magnitudes from the Tycho2 catalogue (Høg et al. 2000) transformed into Johnson system. (B-V) are also from Tycho2. Planet-harboring stars are marked with an asterisk.
As one can see from Table 1, for the majority of stars we get an error which is smaller than 10 K. The consistency of the results derived from the ratios of lines representing different elements is very reassuring. It shows that our 105 calibrations are essentially independent of micro-turbulence, LTE departures, abundances, rotation and other individual properties of stars. We admit though that a small systematic error may exist for below 5000 K where we had only few standard stars.
As was already mentioned, for the first approximation we took accurate temperatures from AAMR96, BLG98 and DB98. The comparison of our final with those derived by AAMR96, BLG98 and DB98 is shown in Fig. 3. As a test of the internal precision of our we investigate the - color relation with the Strömgren index b-y, using our determinations of , and those obtained by other authors. The results are shown in Table 2 where the rms of the linear fit is given for each author's determination, along with our estimate of and using common stars. In each case the scatter of the color relation is significantly improved when adopting our temperatures, though some residual dispersion is still present that can be attributed to the photometric errors, reddening and the intrinsic properties of stars (metallicity, gravity, binarity...) to which the color indices are known to be sensitive. The improvement is particulary spectacular in comparison with EDV93. This proves the high quality of our temperatures and the mediocrity of b-y as a temperature indicator.
Table 2: RMS of a linear regression between and Strömgren b-y using effective temperatures obtained by other authors and in this study with N common stars.
Another point concerns the difference between the zero-point of our temperature scale and that of other authors. Comparing 30 common objects, we find that the AAMR96 scale underestimates temperatures by 45 K near the solar value compared to ours, but apart from that, the deviations are random and no trend with is present. The 45 K offset may arise from the various complications associated with observing the Sun as a star, and/or problems in the models used by AAMR96, like underestimation of convection in the grid of the model atmosphere flux developed by Kurucz. After correcting the AAMR96 zero-point for the 45 K offset, the mean random error of their scale becomes 65 K (where we neglect the error of our own scale which is an order of magnitude less).
The temperatures of BLG98 are also in a good agreement with our
estimates - except for a 48 K offset, no correlation of the
difference with temperature is observed. The mean dispersion is 63 K (for 26 common stars), which is within the errors of the BLG98 scale.
Figure 4: Standard error of the mean versus effective temperature averaged over all available line ratios. | |
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Comparing with DB98: for the 29 stars in common, their temperatures are on average 41 K below ours, and the mean error is 53 K.
Thus, the temperatures derived in AAMR96, BLG98 and DB98 have good precision, though the absolute values are somewhat low relative to the Sun. The reason may be due to the difficulty of photometric measurements of the Sun, as well as indicating some problems in the model atmosphere calculations employed. For example, the Sun's temperatures derived in AAMR96 and DB98 are identical - 5763 K, which is below the nominal value of 5777 K. Besides, the mean temperatures of solar analogue stars (spectral types G2-G3, , and Sun being of type G2.5) derived in these papers are significantly below the solar value: K (AAMR96, 3 stars), K (BLG98, 11 stars) and K (DB98, 7 stars). Our determination for the G2-G3 spectral types is K, based on 12 stars. This demonstrates that a small error (0.8%) affects the zero point of the IRFM method, because when applied to the Sun and the solar type stars, it returns inconsistent results.
We also compared our estimates of with photometric temperatures. EDV93 derived temperatures of 189 nearby field F, G disk dwarfs using the theoretical calibration of temperature versus Strömgren (b-y) photometry (see Table 1). The mean difference between the of Edvardsson et al. (1993) and ours is only -14 K ( 67 K, based on 30 common stars).
To compare our temperatures to Gray (1994), we used his calibration of corrected for metallicity. Our scale is +11 K lower ( 61 K, 24 stars).
Summarizing, we demonstrated that our temperature scale is in excellent agreement with the widely used photometric scales, while both the IRFM method and the method of surface brightness predict too low values for the temperature of the Sun and the solar type stars.
Figure 4 shows the sensitivity of our technique to temperature. Two outliers with errors greater than 20 K are the cold dwarfs HD 28343 and HD 201092, known as flaring stars. For other stars the internal errors range between 3 and 13 K, with a median of 6 K.
The high-precision temperatures were derived for a set of 181 dwarfs, which may serve as temperature standards in the 4000-6150 K range. These temperatures are precise to within 3-13 K (median 6 K) for the major fraction of the sample, except for the two outliers. We demonstrated that the line ratio technique is capable of detecting variations in of a given star as small as 1-5 K. This precision may be enough to detect star spots and Solar-type activity cycles. Of particular interest is the application of this method to testing ambiguous cases of low-mass planet detection, since planets do not cause temperature variations, unlike spots.
The next step will be the adaptation of this method to a wider range of spectral types and for an automatic pipeline analysis of large spectral databases.
Acknowledgements
V.K. wants to thank the staff of Observatoire de Bordeaux for the kind hospitality during his stay there. The authors are also grateful to the anonymous referee for the careful reading of the manuscript and the numerous important remarks that helped to improve the paper.
Table 1F: Program stars. Asterisks indicate stars with planets.