Method to estimate the effective temperatures of late-type giants using line-depth ratios in the wavelength range 0.97–1.32 μm

Abstract

1 INTRODUCTION

Stellar atmospheres are characterized mainly by effective temperature (T_eff), surface gravity (log g) and metallicity ([Fe/H]), of which T_eff has a particularly strong impact on spectra and is often the first parameter to be estimated during spectral analysis. Among many methods of measuring T_eff, we focus on the line-depth ratio (LDR) method in this article. In cool stars (around solar temperature and below), the depths of low-excitation lines of neutral atoms are sensitive to T_eff, while those of high-excitation lines are relatively insensitive (Gray 2008) and hence their ratios are good temperature indicators (Gray & Johanson 1991; Fukue et al. 2015 and references therein). This method has a few advantages: LDRs are not affected by interstellar reddening, and atmospheric parameters other than T_eff are expected to have no, or at least relatively weak, effects on LDRs.

Sasselov & Lester (1990) used LDRs between C i and Si i lines around 1.1 μm to measure the temperatures and reddenings of Cepheids in a relative way. This pioneering work, however, had a limited impact on later applications, due to the small number of standard stars and line pairs. In contrast, previous works on the LDR method mostly used visible spectral ranges (e.g. Kovtyukh et al. 2003, 2006; Kovtyukh 2007). With the development of infrared facilities in recent years, it has become possible to make more comprehensive studies using high-resolution infrared spectra. Recently, Fukue et al. (2015) established the LDR method for H-band spectra, 1.50–1.65 μm. In this wavelength range, however, the small number of low-excitation lines and severe blending between molecular and metal lines make it difficult to find a large number of useful LDR pairs. As a result, the precision of T_eff after taking an average of the values from available pairs is only ∼50 K for each star in the best case, while the LDR method in the visible range can achieve a precision of ∼5 K (Kovtyukh et al. 2006). Here, we investigate Y- and J-band spectra to find the LDR–T_eff relations for the first time. This wavelength range has a number of both low- and high-excitation lines (e.g. Meléndez & Barbuy 1999) and thus one can expect many useful LDR pairs that will enable us to achieve high precision.

2 OBSERVATION

The ten targets listed in Table 1 were selected mainly from Gaia FGK benchmark stars (Blanco-Cuaresma et al. 2014; Jofré et al. 2014; Heiter et al. 2015), in addition to a few prototypes of red giants such as Arcturus. All of our targets are giants with luminosity class III or II and have spectral types between early G and early M (3700 K < T_eff < 5400 K). Their metallicities are between −0.45 and +0.25 dex in [Fe/H] except for Arcturus, −0.52 dex.

We observed these ten stars between 2013 February 23 and 2014 January 23 using near-infrared high-resolution spectrograph WINERED attached to the Nasmyth platform of the 1.3-m Araki Telescope at Koyama Astronomical Observatory of Kyoto Sangyo University in Japan (Ikeda et al. 2016). WINERED can collect spectra covering the wavelength range from 0.90–1.35 μm (z΄, Y and J bands) with a spectral resolution of R ∼ 28 000 with one integration. All of our targets are bright, −2.3 < J < 2.0 mag, and the total integration time with each target within the slit was between 12 and 240 s to achieve a signal-to-noise ratio (S/N) of 100 or higher. We describe the resultant S/N in Section 3.1. For every target star, we also observed a telluric standard star (an A0V star in most cases) for subtracting telluric absorption.

3 DATA ANALYSIS

3.1 Spectra reduction

Basic data reduction was performed automatically using the pipeline software prepared by the WINERED team based on pyraf.¹ The pipeline process includes the standard analysis steps for echelle spectra: bad pixel masking, sky subtraction, flat-fielding, scattered light subtraction, transformation of each two-dimensional echelle image into images with the space and wavelength axes orthogonal to each other for individual orders, spectrum extraction, wavelength calibration and continuum normalization. In addition, it was necessary to correct for time-dependent wavelength shifts as follows. We used ThAr lamp data as the initial wavelength calibration in the pipeline software, but we found that some spectra had wavelength offsets, probably caused by varying ambient temperature.² In this work, such offsets are not critical, because we are not interested in radial velocities, but the adjustment of the wavelength scale of individual frames was necessary to avoid the artificial broadening of the line spread function in a combined spectrum of each target. The relative offsets between individual spectra were measured using almost isolated telluric absorption lines and were removed before combining the spectra. Then, telluric absorption lines in the combined spectrum of each target were removed using the spectrum of a telluric-standard star, except for the 53rd–55th orders, 1.01–1.07 μm, in which almost no significant telluric lines are present. The wavelength scale of each telluric-corrected spectrum was converted to the rest frame of each star using its intrinsic absorption lines. Our wavelengths are in the standard air scale. Finally, we renormalize the spectra using pyraf to adjust the continuum level to unity. In the following discussions, we consider 12 orders listed in Table 2. We ignore other orders (42nd, 49th–51st and 58th–61st), which are around gaps between atmospheric windows. In the analysis to find and calibrate the LDR–T_eff relations, we ignore Arcturus because, according to Fukue et al. (2015), its low metallicity causes offsets from the trends between LDR and effective temperature. Table 2 shows the S/N of each order of each star calculated from the variance of the normalized counts in continuum regions, as an indicator of spectral quality. The spectral resolution was stable and quite constant for all of our spectra. The full width at half-maximum (FWHM) values measured with most stellar lines correspond to λ/Δλ between 26 000 and 30 000. Fig. 1 shows a small range of the spectra obtained. This range has a line pair that shows a clear temperature dependence, as we discuss below.

Figure 1.

Part of the spectra for the 55th order, drawn in order of decreasing effective temperature (from top), except for Arcturus at the bottom, which was not included in establishing the LDR–T_eff relations. The two lines in the graph show a clear dependence on temperature (the high-excitation line on the left and the low-excitation line on the right).

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3.2 Line selection and measurement of line depth

As the candidates of atomic lines to be used for LDRs, we consider the lines identified in our Arcturus spectrum. Here we outline its line identification, but details of the analysis and the entire line list will be published in a forthcoming article (Ikeda et al., in preparation). First, we produced a synthesized spectrum of Arcturus, the parameters of which are assumed to be T_eff = 4275 K, log g = 1.7 and [Fe/H] = −0.4 dex (Smith et al. 2013), using the ATLAS9 atmospheric models and codes given by Kurucz (1993). As the input list of atomic lines for the synthetic spectrum, the Vienna Atomic Line Database (VALD3; Ryabchikova et al. 2015) was used. Secondly, the observed and synthetic spectra were compared closely with each other in order to identify lines with depth larger than ∼1 per cent. Although the S/N of the Arcturus spectrum is higher than 200 at all wavelengths, CN molecular lines often make the identification of shallow lines less certain. In such cases, to confirm the contribution of weak atomic lines around molecular lines, we further compared the observed spectrum with a synthetic spectrum, but with only CN lines included. Thus we obtained approximately 800 atomic lines.

Among the lines identified in Arcturus, here we include relatively isolated neutral lines of Fe, Ti, Si, Cr, Ca, Ni, Mg, Na, Co, Al, Mn and K in our analysis for LDRs. Ion lines and lines of C, N and O were not included (Kovtyukh et al. 2006). Moreover, we excluded lines based on the following criteria: deep lines that appear deeper than ∼0.5 in Arcturus and lines blended with molecular CN lines or other atomic lines, which have different elements and/or excitation potentials. These criteria left 125 and 99 lines in the Y and J bands, respectively. We refer to VALD3 for line parameters such as excitation potential and oscillator strength.

For the 125 and 99 lines we selected, we used a quadratic function to fit three (or four) pixels near the bottom of each line, rather than fitting the entire line profile, e.g. by the Gaussian or the Voigt function (Strassmeier & Schordan 2000). We define |$d_{i}^{(n)}$| as a line depth from the continuum level to the bottom of the fitted function, where i indicates the ID number of the line and (n) indicates the ID number of the star. Weak lines with |$d_{i}^{(n)}<0.02$| are ignored in the following analysis.

3.3 Line pair selection

We calculated the LDR, |$r_{j}^{(n)}=d_{i}^{(n)}/d_{i^{\prime }}^{(n)}$|⁠, for any line pair, j, of approximately 1500 pairs for which the both lines were detected in more than four stars and have excitation potentials separated by more than 1 eV. We treated each order independently, i.e. we did not combine lines in different orders, and divided the depth of the low-excitation line, d_i, by that of the high-excitation line, |$d_{i^{\prime }}$|⁠. For each line pair, j, we plotted the effective temperatures T_eff against the common logarithms of the LDRs, log r_j, and determined the regression line, T_eff = a_jlog r_j + b_j, using the weighted total least-squares method (see Markovsky & Van Huffel 2007 for a review). The weight of a line pair, j, of star (n) was calculated by |$w_{j}^{(n)}=[(\sigma _{y}^{(n)})^{2}+{a_{j}}^{2}(\sigma _{x,j}^{(n)})^{2}]^{-1}$|⁠, where |$\sigma _{x,j}^{(n)}$| and |$\sigma _{y}^{(n)}$| indicate the standard error of |$\log r_{j}^{(n)}$| calculated using the S/N and that of the literature T_eff⁽ⁿ⁾ for each star, respectively. The dispersion around each regression line is defined as σ_j². The effective temperature and its error for each star based on each relation are given as |$T_{j}^{(n)}\pm \Delta T_{j}^{(n)}$|⁠, for which the error was determined using the variance–covariance matrix of coefficients (a_j, b_j) of the regression line and the error of |$\log r_{j}^{(n)}$|⁠. In the subsequent analysis, about 900 line pairs that have σ_j > 150 K were not included. Moreover, 21 line pairs that have a_j > 0 were excluded. A large fraction of the rejected pairs include a weak iron line at 10350.77 Å (6.145 eV) in the Y band. This line may be blended by other line(s), which causes the unexpected dependence of the LDR on effective temperature, though the presence of such blended lines is not clear in the available line lists. The rest of the line pairs with a_j > 0 have Ti i lines for both low- and high-excitation potentials.³ LDR–T_eff relations for the majority of Ti i–Ti i pairs are not tight enough to be selected and their slopes can be negative, positive or not well determined. The reason for the opposite slopes for a small number of Ti i–Ti i pairs is unclear.

There is a large number of possible combinations of line pairs and we selected a set of pairs as follows. One condition is that each line is used only once, i.e. it is not included in more than one line pair. This ensures that statistical errors in |$r_{j}^{(n)}$| values in different LDR–T_eff relations are independent of each other and makes it easy to calculate the statistical errors of combined effective temperatures (T_LDR in Section 4). In fact, the final set of relations we discuss in Section 4 makes use of 162 lines, which is about 70 per cent of the selected lines. This condition for the line-pair selection has no large impact on the statistical error, which would be better only by ∼15 per cent even if we used all the pre-selected pairs.

The basic idea of the process is to select line pairs that meet the following conditions as much as possible: (i) high precision in reproducing the effective temperatures of our sample stars and (ii) small difference in wavelength between the two lines of each line pair. With the latter condition fulfilled, the possibility of other instruments detecting both lines of a line pair in the same echelle order becomes high and the error in the LDR introduced in the continuum normalization is expected to be smaller.

4 RESULTS

We applied the procedure described above to our 224 nodes (i.e. lines) and 603 edges (i.e. line pairs). By changing the coefficient e from 0 to 2 K Å^{− 1}, we searched for the optimized matching, M_k(e), that gives the smallest E at each e value and observed how e affects the solutions. Fig. 2 plots the values of E_T and E_λ for M_k(e) with varying e. By increasing e, the weight of E_λ increases relative to E_T in the evaluation function and then the optimal matching and the values of E_T and E_λ change. At around e = 0.5 K Å^{− 1}, E_T is only slightly larger compared with the case with e = 0 K Å^{− 1} for most orders (the exception being the order 48), which optimizes the precision in the redetemined temperature by ignoring the difference in wavelength, while E_λ improves to some extent by changing e from 0 to 0.5 K Å^{− 1}. The number of line pairs in the 48th order is only four and, with e = 0.5 K Å^{− 1}, the wavelength separations of the individual pairs become significantly small, with only a small increase in E_T. We selected e = 0.5 K Å^{− 1} and Table 3 lists the resultant values of the evaluation functions and other parameters in our analysis.

Figure 2.

The dependence of E_T(M_k(e)) and E_λ(M_k(e)) on the parameter e in equation (1). The evaluation functions are defined in the text. Variations of E_T(M_k(e)) (top row) and E_λ(M_k(e)) (bottom row) are presented for a group of echelle orders in each band (52nd–57th in the Y band on the left and 43rd–48th in the J band on the right). Each solid line represents the evaluation function and the vertical dashed line represents the value e = 0.5 K Å^{− 1} that we used for the final solutions. Legends show which colour corresponds to which echelle order. [The colour version of this figure is available in the online journal.]

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We obtained 47 LDR–T_eff relations in the Y band and 34 in the J band, with e = 0.5 K Å^{− 1}. Tables 4 and 5 list the line pairs and relations and Fig. 3 shows some examples of these relations. We redetermined the effective temperatures using our final relations (T_LDR in Table 6) and compared them with T_eff in the literature in Fig. 4. The difference, T_LDR − T_eff, has no clear dependence on T_eff or [Fe/H], suggesting that this LDR method is effective across the parameter range covered by the nine calibrating stars, i.e. 3700 < T_eff < 5400 K and −0.5 < [Fe/H] < +0.3 dex. In order to evaluate the power of the LDR method, considering all orders available, we consider the evaluation function of equation (1) but including all orders and line pairs, i.e. we use T_LDR⁽ⁿ⁾ instead of the temperatures for an individual order, |$T_{M_{k(e)}}^{(n)}$|⁠, and consider the wavelength differences of all line pairs. Thus the calculated evaluation functions are |$E_{T}^{\mathrm{all}}=28.6\, \mathrm{K}$| and |$E_{\lambda }^{\mathrm{all}}=68.0\, \mathrm{{\rm \mathring{\rm A} }}$|⁠. These values are, for example, smaller than those⁴ for the result by Fukue et al. (2015), who used nine line pairs in the H band and obtained |$E_{T}^{\mathrm{IRCS}}=85.5\, \mathrm{K}$| and |$E_{\lambda }^{\mathrm{IRCS}}=295.6\, \mathrm{{\rm \mathring{\rm A} }}$|⁠.

Figure 3.

Examples of LDR–T_eff relations. The ID of the line pair is given in each panel. Arcturus, indicated by a green open square in each panel, was not used for the relation that was obtained; other objects are indicated by blue filled circles. Plots for all 81 relations are available as online material – see the Supporting Information.

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Figure 4.

Differences between the redetermined effective temperatures T_LDR from our LDR method and the literature temperatures T_eff plotted against T_eff (left panel) and [Fe/H] (right panel). Symbols are the same as in Fig. 3. The size of the vertical error bars is dominated by the error in the literature temperature, except for high-temperature stars (ε Leo and κ Gem) and Arcturus.

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