2.1. Determination of Equivalent Widths

The quantification of equivalent widths was done in an automatic manner. For this purpose we have developed a code that replaces the true continuum by a pseudo-continuum through the interpolation of a straight line that connects the peaks on both sides of an absorption line (see Figure 1).

Fig. 1 Maximum and minimun points (red squares) that allow us to derive the equivalent widths for IRAS 08281 - 4850 (F0I) and IRAS 17332 - 2215 (K2I) using an automatic code. The location of different absorption lines selected are labeled by continuous lines. The color figure can be viewed online. 

The equivalent width is then defined as the effective area occupied between the two maximum interpolated Wj= Σj=1nIc-IjIc∆λ, where ∆λ is wavelength interval (or its dispersion). Tables 1 and 2 show the quantitative measures of 9 equivalent widths of absorption lines selected in this study.

We can compare the measurements of equivalent widths from the automatic code with those done manually with the IRAF code. We use the quantifiable parameters of IRAS 01919 + 0373, IRAS 08281 4850, IRAS 22223 + 4327 with spectral types A0, F0 and G0 respectively. From Figure 2 we note that for weaker absorption lines (i.e. with low measures and intermediate equivalent widths) the values are in good agreement among themselves, while for stronger lines the values show slight systematic differences between them, which slightly increase the error. The outliers obtained with this method are usually due to poorly measured equivalent widths, caused by a poor maximum point determination.

Fig. 2 Correlation between the equivalent widths measured with the automatic code and those taken from IRAF code. Three objects IRAS 01919 + 0373, IRAS 08281 4850, IRAS 22223 + 4327, with spectral types A0, F0 and G0, have been used to compare the results. 

2.2. MK Criteria

Our atmospheric parameters were estimated from features used by the MK system.

To determine the effective temperature we have used the equivalent width of the calcium line Ca IIK at λ3933 Å (warm stars) and the G-band at λ4302 Å (cool stars). The Ca IIK feature grows dramatically in strength from A-type toward late types (~F8); for cooler spectral types the equivalent widths remain flat. Other features, as the Ca I at λ4226 ̊A and the Mn I at λ4030 Å, are not useful to estimate the temperature in warm stars.

For G-type stars, the G-band characteristically dominates over other features. This feature increases in strength until about K2 and then decreases in intensity. Another feature, the MgI triplet at λ5167-72 Å shows some sensitivity to temperature for cold objects.

For the determination of surface gravity we employ ionized lines as criteria, the (λ4172-79 Å and λ4395-4400 Å blends, Sr II at λ4077 Å and Mg II at λ4481 Å). It is also possible to use the neutral oxygen triplet (λ7771-5 Å) located in the near IR-region. In warm stars, however, only the λ4172-79 Å blend of Fe II and Ti II is gravity sensitive, while the Sr II at λ4077 Å and the Ca I at λ4226 Å lines are sensitive to gravity in cold stars.

In order to obtain the stellar metallicity we used only the absorption lines of neutral iron, i.e. Fe I (λ4063 Å), the Fe I (λ4271 Å) blend and Fe I (λ4383 Å). We discarded any ionized iron lines because of their expected dependence on logg. For warm stars, we used as metallicity indicator the sum of the iron lines Fe I (λ4271 Å blend + λ4383 Å), while the Fe I lines (λ4063 Å and λ4383 Å) were employed for cool stars.

The lines of NaID at λ5889-95 Å and OIT at λ7771-5 Å were used as probable indicators for the determination of the stellar distance. The interstellar component of the Na ID lines at λ5889-95 Å showed sensitivity to luminosity in young stars. However, in evolved stars (as PAGB stars) both lines are affected by circumstellar material and therefore do not show a dependence on luminosity. The O IT lines at λ7771-5 Å, on the other hand, do show sensitivity to luminosity. In fact, ^{Arellano Ferro et al.(2003)} found accurate spectroscopic calibrations between visual absolute magnitudes and the OIT lines for a sample of 27 calibrator stars with spectral types A to G.

Because of the spectral range limitation in the near infrared region, the lines of the hydrogen Paschen series, and the oxygen and calcium triplets are very rare in the total sample. The equivalent widths of all features are presented in Tables 1 and 2, respectively.

2.3. Error in the Equivalent Widths

An accurate determination of systematic and random errors of the equivalent widths is not trivial, since these are a function of the magnitude, spectral type, the S/N ratio and the pseudo-continuum position. We also need common stars with the same spectral types and several measures of their stellar spectra. This sample has limitations for objects of the same spectral types and also for those with scarce measurements of equivalent widths, so that it is impossible to carry out a reliable statistics.

In this section we can estimate an approximation between the errors of the equivalent widths of the selected absorption lines and the spectral types. For this purpose, the spectral types were replaced by numerical values, as in the following sequence: A0 = 30, F0 = 40, G0 = 50 and K0 = 60, respectively. The intermediate values are taken between two successive classes. In view of the difficulties presented by the observational data (mentioned above), we decided to correlate the equivalent widths determined through the automatic code with those obtained from the IRAF code for both samples as shown in Figure 2.

A dispersion, σ_EW, is obtained for each spectral type (or an average σ_EW, if the spectral type is repeated), which varies from 0.02 to 0.47 for the warm stars and from 0.15 to 0.35 for the cold stars. In Figure 3 we observe that the dispersion shows a tendency to increase with increasing spectral type for both warm and cold stars. This dispersion results in an error in the effective temperature, ∆T_eff, such that using equation 1 leads to a variation between 5 K and 68 K, and using equations 2 and 3 from 63K to 122K, respectively.

Fig. 3 Relation between the stellar atmospheric parameters as a function of equivalent widths for warm stars (top panel) and cold stars (bottom panel). The empty squares represent stars left out of the best fit. 

With these arguments we can infer that the new spectroscopic calibrations for effective temperature, gravity and metallicity are not affected by the dispersion of the equivalent withs.

2.4. Sample for Calibration

Table 3 shows the PAGB stars that have been studied and reported in the literature. This table contains the number IRAS, the spectral type, the stellar atmospheric parameters obtained from different sources and their respective references. The stellar parameters were obtanied by different authors using spectroscopic methods.

We can observe that there are a total of 21 stars considered to be warm-PAGB stars and a very small number, only 5 objects, considered to be cold-PAGB stars. In spite of having a small number of stars as calibrators, it is possible to obtain a rapid and accurate determination of the fundamental parameters (effective temperature, surface gravity and metallicity) using only suitable spectral criteria, avoiding photometric indices which are often distorted by poorly known interstellar and circumstellar reddening.

In the recent past, two papers that involve photometric calibrations (Strömgren and 2MASS photometry) and that allow to estimate stellar parameters for a group of post-AGB and RV Tauri stars were published by ^{Arellano Ferro et al. (2010)} and ^{Molina (2012)}.

2.5. Polynomial Fitting

The stellar atmospheric parameters can be determined by fitting a series of polinomials whose independent variables are the equivalent widths. Our goal is to analyze the actual dependence of the stellar parameters on one or two quantifiable features. The mathematical representation of the polynomial, in general, has the form

where V is any of the three stellar parameters (T_eff, log g and [Fe/H]), a_ij are the coefficients to determine and x and y are the independent variables. When the number of independent variables is greater than one, we used the method adopted by ^{Stock & Stock (1999)}, who developed a quantitative method to obtain stellar physical parameters such as absolute magnitude, intrinsic color, and a metallicity index using the equivalent widths of absortion features in stellar spectra by means of polynomials and a consistent algorithm (^{Molina & Stock 2004}).

In order to determine the best coefficients we employ an algorithm based on least squares. This algorithm performs an initial fitting and removes those values of residuals greater than 2σ. The error of each coefficient is obtained from

where S(i, j) the diagonal matriz and σ² is the mean square error.

3.1. Teff Calibration

For the determination of the effective temperature in warm-PAGB stars we use the equivalent widths of the Ca IIK at λ3933 Å. This line has been considered in the MK system as an indicator of temperature in warm stars (Gray & Corbally 2008). Particularly, for stars with temperatures between (6000 ≤ T_eff ≤ 8000K), the equivalent widths show sensitivity to effective temperature. A code based on least squares that relates equivalent widths and effective temperatures taken from the literature (T_eff^ref) leads to the following relationship

This calibration is valid for a range of equivalent widths between 0.76 ≤ T_eff ≤ 13.15 Å. The standard deviation derived from equation (1) is ±91 K. Four stars were left out of the fit, i.e. IRAS 08143 - 4406, IRAS 08281 - 4850, IRAS 14325 - 6428 and IRAS 22223 + 4327. The stellar temperatures estimated by ^{De Smedt et al. (2016)} for IRAS08281- 4850, IRAS 14325 - 6428 and IRAS 22223 + 4327 are 7875 K, 8000 K and 6500 K, respectively and that estimated by ^{Reyniers et al. (2004)} for IRAS08143- 4406 is 7150 K, while the fit of equation (1) leads to values of 7674 K, 7211 K, 6809 K and 6856 K, respectively.

For late-PAGB stars (4500 ≤ T_eff ≤ 5500 K), it is possible to determine the effective temperature from the G-band at λ4302 Å. In spite of only 5 stars being present in the fit, it is possible to determine the effective temperature applying a linear fit

Equation (2) is valid for a range of equivalent widths between 1.76 ≤ T_eff ≤ 4.08 Å and the standard deviation achieved is ±207K. Two objects are left out of this relationship, IRAS 01259 + 6823 (5000 K), IRAS 22223 + 4327 (4500 K); the fit for both objects gives the same temperature value of 4788 K.

The stellar temperature for identified cold PAGB stars can be increased by using the resonance Ca I (λ4226 Å) line. This line is sensitive to temperature, since it grows gradually from G-type to early K-type stars, being stronger in stars of mid-K type. A linear relationship can be obtained by adjusting the temperature and the Ca I equivalent widths for four calibrating stars, this is

where the standard deviation reaches a value of ±175 K and the validation range for the equivalent widths can be found between 0.40 Å and 2.31 Å and for the temperatures between λ4550 Å and λ5200 Å. The results for the effective temperatures estimated by equations (1), (2) and (3) are shown in the third and fourth column of Tables 4 and 5. At the top of Figure 4, we note the dependence of the Ca IIK-line and the G-band on the effective temperature (see left and right panels).

Fig. 4 Trend of the dispersion σEW with the spectral type. We see that the dispersion tends to increase with the increase of the spectral type for both warm and cold stars. 

3.2. Log g Calibration

For warm stars, we can estimate the surface gravity using the Fe,TiII blend at λ4172-9 Å. This blend is constituted mainly by ionized lines of Fe and Ti and has been considered as an indicator of luminosity in A-F type stars. A linear fit leads to the following relationship

The range of validation of this calibration for surface gravity spans 0.50 ≤log g ≤ 1.40, while the equivalent widths of the ionized line vary between 0.50 ≤ FeTi II ≤ 3.90 Å. The standard desviation has a value of σ=±0.21. Two stars fall out of the fit of equation (4), i.e. IRAS 07134 + 1005 and IRAS 18025 - 3906. According to the spectral types (or effective temperature) of IRAS 07134 + 1005 and IRAS 18025 - 3906 their equivalent widths would be expected to be slightly greater than 4 Å.

We can extend the range of surface gravity to higher values using the O I triplet lines. Due to the limitations of the spectral range to the near infrared region, the number of O I triplet lines is scarce. Their values are not reported in Table 4, and we will only show the functional relationship:

where the range of gravity varies from 1.00 to 2.20 and the equivalent widths vary from 0.07 to 1.95 Å. The standard desviation has a value of σ=±0.35.

The surface gravity of cold stars is estimated using the Sr II-line. This line has been considered as the principal luminosity discriminator for cool stars in MK classification. Unfortunately the functional relationship is built with only 3 stars and has the following form

The range of validation of this calibration in surface gravity spans 1.00 ≤log g ≤ 1.50, while the equivalent widths of the ionized line vary between 0.77 ≤ Sr II ≤ 2.95 Å. The standard desviation has a value of σ=±0.20. We can also estimate the gravity for additional cold PAGB stars: 13203−5917, 16494−3930, 17317−2743 and 17388−2203 when recovering the equivalent widths of the Sr II line from the Mg II line. An error of ±0.25 is introduced by this estimation.

The results for the surface gravity estimated by equations (4) and (6) are shown in the fifth and sixth columns of Tables 4 and 5. In the middle of Figure 4, we see the dependence of the Fe,TiII blend and the Sr II on surface gravity (see left and right panels).

3.3. [Fe/H] Calibration

For the calibration of metallicity we used only neutral Fe lines. For warm stars, we use the sum of Fe I (λ4271 Å + λ4383 Å). The best fit that recovers the metallicity is generated by a polinomial of the form

The range of validation of this calibration on metallicity covers −0.09 ≤ [Fe/H] ≤ −1.00 dex, while the equivalent widths of Fe lines vary between 0.34 ≤ Fe I ≤ 4.40 Å. The standard desviation for this relationship is ±0.19dex. Two outliers are present in this fitting; IRAS 19386 + 0155 of very low metallicity (−1.00 dex) and IRAS 20572 + 4919 of solar metallicity (−0.01 dex).

For cold stars, we employ the Fe I lines at λ4063 Å and λ4383 Å. Of the 17 cold-PAGB stars only 5 objects have identified stellar parameters. For the Fe I (λ4353 Å) line five objects are available for the calibration. The best fitting that recovers the metallicity within a range of −0.09 ≤ [Fe/H] ≤ −0.80 dex, involves a linear polynomial for Fe I line at λ4383 Å, that is,

where the equivalent widths vary between 0.48 and 1.94 Å and the standard desviation has a value of σ=±0.30 dex.

On the contrary, the best fitting for the Fe I line at λ4063 Å has the form

The equivalent widths vary between 0.48 and 2.86 Å and the standard desviation has a value of σ=±0.30 dex.

The results for the metallicity estimated by equations (7) and (8) are shown in the sixth and seventh columns of Tables 4 and 5. At the bottom of Figure 4 we observe the dependence of the Fe I lines on metallicity (see left and right panels).