3 Numerical methods

3.1 Model atmosphere

A model atmosphere was obtained firstly with the ATLAS9 program (Kurucz 1993) adapted for Sun OS by M. Lemke. According to numerical simulation of stellar convection (Abbett et al. 1997) we have selected l/H = 1.5, and for models we put $v_{\mathrm{turb}}=2.0\,\mathrm{km}~\mathrm{s}^{-1}$.

Second, we applied the POSMARCS grid of models (Plez 1997) described in some details by Bessell et al. (1998) and based on SOSMARCS models (Plez et al. 1992). The microturbulent velocity was chosen as $v_{\mathrm{turb}}=2.0\,\mathrm{km}~\mathrm{s}^{-1}$.

3.2 Synthetic spectra

The synthetic spectra are calculated by the program SYNSPEC36 (Hubeny et al. 1994). Several modifications of this program have been made to adapt the program to spectrum synthesis of cool stars. The most important modifications are the inclusion of atoms with $30<Z\leq 90$ and the inclusion of selected transitions of some diatomic molecules.

The polynomial partition function approximations for atoms with $30<Z\leq 90$ are adopted from Irwin (1981). The molecular constants are taken from Huber & Herzberg (1979), while the line list is taken from Kurucz's CD-ROM (1993). Atomic line data for the spectrum synthesis are adopted from VALD (Piskunov et al. 1995). The Hubeny's program ROTIN3 computes rotational and instrumental convolution.

3.3 Optimization

Two steps (computation of model atmosphere and synthetic spectrum) are repeated to match the observed spectrum. A non-linear method optimizes the model parameters and/or synthetic spectrum parameters. To avoid numerical problems during the calculation of gradient we have chosen a simplex method.

We used two different methods for the calculation of the optimization criterion (see below).

3.3.1 Direct comparison

  The computed synthetic flux is convolved with the rotational and instrumental kernel function. The final synthetic spectrum is obtained in a denser wavelength grid than the observed one (because of more accurate computation of the convolution), so a spline interpolation is applied to obtain a uniform wavelength grid. The optimization criterion is

$$\sigma^2=\sum_n \left(H_{n}^{\mathrm{obs}}-H_{n}^{\mathrm{syn}}\right)^2$$ (1)

where H is the line profile and subscripts $\mathrm{obs}$ and ${\mathrm{syn}}$ denote observed and synthetic spectrum, respectively.

3.3.2 Comparison applying Fourier transform

  Since the rotational velocity $v\sin i$ and the so called macroturbulent velocity are not important quantities for stellar atmosphere, we applied the Fourier transform method for adopting the basic parameters of stellar atmosphere. Lowering the number of independent variables significantly decreases the computer time spent for each fitting, and does not require knowledge of the detailed shape of the instrumental profile.

An ideal observed spectrum $H_{\mathrm{ideal}}$ for slowly rotating stars is the convolution of the flux emergent from the star $H_{\mathrm{em}}$ and the rotational profile $I_{\mathrm{rot}}$ (merged with the profile of the macroturbulent velocity); let us write according to Gray (1992)

$$H_{\mathrm{ideal}}=H_{\mathrm{em}}\ast I_{\mathrm{rot}}.$$ (2)

Likewise, the final observed spectrum $H_{\mathrm{inst}}$ is a convolution of an ideal observed spectrum and instrumental profile $I_{\mathrm{inst}}$

$$H_{\mathrm{obs}}=H_{\mathrm{ideal}}\ast I_{\mathrm{inst}}.$$ (3)

Applying known features of the convolution we can now write

 

$$H_{\mathrm{obs}}=H_{\mathrm{em}}\ast I,$$ (4)

where I is the convolution of rotational and instrumental profile

$$I=I_{\mathrm{rot}}\ast I_{\mathrm{inst}}.$$ (5)

Fourier transform of Eq. (4) gives

 

$${\cal H_{\mathrm{obs}}}={\cal H_{\mathrm{em}}}{\cal I}$$ (6)

where ${\cal H_{\mathrm{obs}}}$, ${\cal H_{\mathrm{em}}}$ and ${\cal I}$ denote the Fourier transform of the corresponding quantities. From this equation we can now determine the instrumental profile I as

 

$$I\equiv {\cal F}\left({\cal I}\right) ={\cal F}\left({\cal H_{\mathrm{obs}}}/{\cal H_{\mathrm{em}}}\right)$$ (7)

where ${\cal F}$ denotes the Fourier transform. A typical result of such a calculation is a function which has non-zero values only in a very small interval. This result is demonstrated in Fig. 1 and can be used for the optimization, as described below.

  

Figure 1: A sample of the Fourier transformation of the convolution of tentative instrumental and rotational profiles. The solid line is a part of that Fourier transformation (for $\beta$ Gem) computed with optimum parameters (given in Table 2) and the dashed lin e is a part of that Fourier transform with temperature shifted about 500 K. The peak (around n=0) represents the Fourier transform of the convolution of the rotational and instrumental profile. For optimum parameters this peak is higher. Furthermore, beginning from a certain point (in this case $n\raisebox{-1pt}[0pt]{$\stackrel{\gt}{\sim}$\space }10$) values of ${\cal I}$ computed with shifted temperature are higher than values computed with correct temperature. If we could compute an accurate synthetic spectrum, then the values outside the peak would be equal to zero

The computed synthetic spectrum is spline interpolated to $n_{\rm obs}$ wavelength points of the observed spectrum. Then, FFT is performed on both observed and synthetic spectrum to obtain ${\cal H_{\mathrm{obs}}}$ and ${\cal H_{\mathrm{em}}}$,where ${\cal H_{\mathrm{em}}}$ now denotes the Fourier transform of the synthetic spectrum. Equation (6) is consequently used to compute ${\cal I}$, the Fourier transform of convolution of instrumental and rotational profiles. Furthermore, backward FFT is performed on derived profile ${\cal I}$, applying Eq. (7). The final optimization criterion is computed in the form of  

$$\sigma^2=\sum_{n=n_{\rm inst}}^{n_{\rm obs}-n_{\rm inst}}I_n^2,$$ (8)

where $2\cdot n_{\rm inst}$ is the number of points where the profile is significantly non-zero. Globally, the tentative instrumental profile I is computed from Eq. (7), where ${\cal H_{\mathrm{em}}}$ is substituted by the Fourier transform of the synthetic spectrum and ${\cal H_{\mathrm{obs}}}$ is the Fourier transform of the observed spectrum. Different values of $\sigma^2$ can be obtained by varying the parameters of the synthetic spectrum. The parameters of spectra with the lowest $\sigma^2$ are close to genuine atmosphere parameters of the star.

To indicate the quality of the fit, we used the quantity $\overline{\sigma}$, defined as

$$\overline{\sigma}=\sqrt{\frac{\sigma^2}{n_{\rm obs}}}$$ (9)

where $n_{\rm obs}$ is number of points of the observed spectrum.