4 Calibration of theoretical spectra

 

4.1 Correction procedure and conservation of the original differential properties

 The need to re-calibrate the theoretical SEDs was clearly demonstrated in the previous work, by comparing (i) the model colors obtained from the original synthetic spectra with the empirical temperature-color calibrations, and (ii) the model colors originating from the different basic libraries between themselves. In Fig. 6 of LCB97, discrepancies as large as $\sim$ 0.5 mag can be found for instance in U-B and B-V for $T_{\mathrm{eff}}$ < 3500 K. A correction procedure was developed in order to provide color-calibrated theoretical fluxes over a large wavelength range, (typically from U to L), while preserving the color differences due to gravity and metallicity variations given by the different original grids. This method was based on the definition of correction functions at a given effective temperature, $\Phi_{\lambda} (T)$, obtained from the ratio of the corresponding empirical and theoretical pseudo-continua at each wavelength. The pseudo-continua in turn were calculated from the colors and, hence, the monochromatic fluxes within each wavelength band as black-bodies with smoothed color temperatures varying with wavelength, $T_{\rm c}(\lambda)$: 

$$pc_{\lambda}(T_{\rm eff}) \propto B_{\lambda}(T\rm _c(\lambda)).$$ (5)

A corrected spectrum can therefore be simply computed by multiplying the original spectrum for a given set of stellar parameters ($T_{\mathrm{eff}}$,$\log g$, [M/H]) with the corresponding correction function at the same effective temperature. A scaling factor, $\xi(T,\log g,\chi)$, is finally applied to the corrected spectrum in order to conserve the bolometric flux (Eq. 9). As the correction function is, by definition, a factor which depends on the effective temperature only, this simple algorithm also preserves, to first order, the original differential grid properties implied by metallicity and/or surface gravity changes, and thus satisfies the basic requirement imposed on the correction procedure.

While this is true for the monochromatic magnitude differences - the correction function becomes an additive constant on a logarithmic scale -, this condition cannot be fully achieved for the (broad-band) colors, which represent heterochromatic measures of the flux. Yet, if we consider an original spectrum for a given set of stellar parameters, $S^{\circ}_{\lambda}(T,\log g_{1},\chi_{1})$, the magnitude m_i in filter i is:

$$\begin{eqnarray} \lefteqn{m_{i}^{\circ}(T,\log g_{1},\chi_{1}) = }\nonumber \\ ... ...,\log g_{1},\chi_{1})P^{i}_{\lambda} \, {\rm d}\lambda + K_{i}\, ,\end{eqnarray}$$ (6)

where $P^{i}_{\lambda}$ is the normalized transmission function of the filter i defined between $\lambda_{m_{i}}$ and $\lambda_{M_{i}}$.For different values of the gravity and the chemical composition at the same $T_{\mathrm{eff}}$, we have

$$\begin{eqnarray} \lefteqn{m_{i}^{\circ}(T,\log g_{2},\chi_{2}) = }\nonumber \\ ... ...,\log g_{2},\chi_{2})P^{i}_{\lambda} \, {\rm d}\lambda + K_{i}\, .\end{eqnarray}$$ (7)

The color difference $\Delta c_{ij}^\circ$ due to the variations in $\log g$and $\chi$ are then given by

$$\begin{eqnarray} \Delta c_{ij}^{\circ}&=-2.5\log & \left( \frac{\displaystyle \i... ...a}(T,\log g_{2},\chi_{2})P^{i}_{\lambda} {\rm d}\lambda} \right).\end{eqnarray}$$ (8)

Since for the corrected spectra (LCB97),  

$$S^{\rm c}_{\lambda}(T,\log g,\chi) = \Phi_{\lambda} (T) \cdot \xi(T,\log g,\chi) \cdot S^{\circ}_{\lambda}(T,\log g,\chi),$$ (9)

we have, after elimination of $\xi$ in the color term:  

$$\Delta c_{ij}^{\rm c} =-2.5\log \left( \frac{\displaystyle ... ...T,\log g_{2},\chi_{2}) P^{i}_{\lambda} {\rm d}\lambda} \right).$$ (10)

Thus the differential colors are rigorously preserved by the correction procedure (i.e. $\Delta c_{ij}^{\circ} = \Delta c_{ij}^{\rm c}$) if $\Phi_{\lambda} (T)$ is constant between $\lambda_{m}$ and $\lambda_{M}$. In practice, these rather severe constraints are well matched - and the corresponding $\delta(\Delta c_{ij})$ are small - if $\Phi_{\lambda} (T)$ varies slowly across the passbands or, equivalently, as long as the filters are not too wide.

Figures 18 and 19 of LCB97 show, in fact, that the color residuals are negligible over the whole ranges of UBVRIJHKL colors and model parameters. Significant residuals ($\sim$ 0.05 mag) are found in R-I at the lowest temperatures because of the more significant variations of the correction functions inside the long-tailed R filter.

4.2 Correction of M dwarf model spectra

 In Fig. 4, we compare a sequence of theoretical colors computed from the dwarf model SEDs to the empirical colors. While the hottest models (K95) match quite well the observed sequences - except for $T_{\mathrm{eff}}$ < 4000 $\sim$ 3750 K - the coolest ones (AH95) exhibit more serious discrepancies, in particular in the blue-optical colors. The Extended models (open squares) provide unrealistic U-B and B-V colors, differing by as much as 2 $\sim$ 3 mag relative to the NextGen version models (crosses). At longer wavelengths, both these new and old M dwarf model generations seem more realistic, although the infrared colors appear systematically too blue probably due to an incomplete $\rm H_{2}O$ opacity list at solar-metallicity used in the calculations (Allard, private communication).

  

Figure 4: Colors of original models of dwarfs compared to our empirical $T_{\mathrm{eff}}$-color relations (solid lines). The ``Extended'' models (open squares) exhibit larger deviations (except in H-K) and unrealistic U-B and B-V colors compared to the ``NextGen'' (labelled ``NG'') models (crosses)

In order to provide more realistic (broad-band) colors for M dwarfs, we have applied the correction method described in Sect. 4.1, using the pseudo-continua calculated from the updated empirical temperature-color relationships between 2000 K and 11500 K. However, in the range 2000 K $\sim$ 4500 K we have to distinguish between the ``dwarf'' and the ``giant'' correction functions derived from model spectra originating from different grids (AH95 and ``B+F'', respectively). This was done by fixing the lower limit of $\log g$ for a ``dwarf'' model to 3.0 dex. Thus, all the spectra with $\log g$ lower than or equal to 2.5 and $T_{\mathrm{eff}}$ less than 4500 K are corrected according to the giant empirical color sequences, while the others are calibrated from the dwarf sequences. For [M/H] = 0, we also defined corrections functions to be applied uniquely to the NextGen models, independently of those computed for the Extended models used at other metallicities. In Fig. 5 we compare the corrected model colors to the $T_{\mathrm{eff}}$-color calibrations: most of the theoretical colors now match very well the empirical relations. For the largest original deviations found in U-B and B-V below 3000 K, important discrepancies still remain (more than 1 mag for the Extended models), but the corrected colors should nevertheless provide more reliable values, in particular those predicted by the NextGen models.

As for the original model spectra, UBVRIJHKLM corrected model colors and bolometric corrections have been synthetised for the whole range of parameters provided by the complete library. Color grids are given in electronic tables accompanying this paper.

  

Figure 5: Same as Fig. 4 but for the corrected models. The corrections provide a perfect match of the empirical colors over the whole range of $T_{\mathrm{eff}}$, except in U-B and B-V below 2500 K, where significant discrepancies persist

Because the corrections are so substantial, the question which naturally arises is how well preserved are the original differential colors for the coolest M dwarfs. We have computed, for these models, the residual color differences between corrected and original model colors, $\delta(\Delta c_{ij}) = (\Delta c_{ij}^{\rm c} - \Delta c_{ij}^{\circ})$. For metal-content variations the results are presented in Fig. 6, where $\delta(\Delta c_{ij})$ is plotted as a function of [M/H]. At low temperatures (< 2200 K) and low metallicities (< -2.0), differences as large as $\sim$ 2 mag are reached for U-B and B-V! For the other colors the residuals are smaller, but typical values of order $\sim$ 0.2 mag still remain for the coolest and the most metal-deficient models. Clearly, the original grid properties are not conserved for these models.

  

Figure 6: Color-difference residuals between corrected and original models as functions of [Fe/H] for some of the M dwarf models. The symbols indicate the difference between the color excess of a corrected model at a given metallicity and the color excess of the corresponding original model, having the same $T_{\mathrm{eff}}$ and $\log g$ (as indicated on the bottom-left panel). The color excesses are calculated as: $\Delta c_{ij} = c_{ij}(\chi) - c_{ij}(\chi_{\odot})$

The main reason behind these deviations is the strong variation of the correction factors within the spectral range covered by broad-band filters, as discussed in Sect. 4.1. Figure 7 shows the correction functions computed for some of the coolest models, compared to the respective positions of the different passbands, and their effect on a cool dwarf model spectrum (bottom panel). As previously, the functions are all normalized at 817 nm in the I band (LCB97) and plotted on a logarithmic scale in order to emphasize the relative differences. As we can see for 2000 K (solid line) and 2500 K (short-dashed line), the correction functions vary considerably across the UBV filters; e.g., the 2000 K-function changes by a factor $\sim$ 1000 between 400 and 600 nm! These steep gradients inevitably degrade the resulting differential colors. Significant fluctuations of correction factors are also present in the JHK bands, accounting for the significant residuals seen in these differential colors. For the giants at 2500 K (thick line), the largest fluctuations appear in R and I, leading to the deviations found in LCB97 ($\sim$ 0.05) at this temperature.

  

Figure 7: Top panel. The correction functions applied to the M star models: for dwarf spectrum at 2000 K (thin solid line), 2500 K (long-dashed line), 3500 K (short-dashed line), and for giant spectrum at 2500 K (thick solid line). The different filters are also shown. Note the strong variations of these functions in the UBV bands for the coolest dwarf models (2000 K and 2500 K) which affect the differential colors. Bottom panel. A corrected M dwarf spectrum (solid line) compared to the original one (dotted-line). An arbitrary shift has been applied for clarity

4.3 Shortcomings of the correction algorithm

 In order to investigate correction methods which better preserve the differential properties, we developed a procedure which avoids the use of functions applied to the spectra as multiplicative factors. This was done by introducing directly, as an input to the correction algorithm, the color differences, $\Delta c_{ij}(T,\Delta \log g,\Delta \chi)$, that we want to preserve from the original models. Generalizing the definitions of the non-solar $T_{\mathrm{eff}}$-color calibrations (Eq. 4), we define the semi-empirical colors of each model in the grid, $\stackrel{\sim}{c_{ij}} (T,\log g,\chi)$, by adding to the empirical colors given in Table 1 the theoretical color differences due to changes in metallicity,$\Delta \chi$, and surface gravity, $\Delta \log g$:

$$\begin{eqnarray} \stackrel{\sim}{c_{ij}} (T,\log g,\chi) & = & c_{ij}^{\rm emp}... ...\nonumber \\ & + & \Delta c_{ij}(T,\Delta \log g,\Delta \chi)\,.\end{eqnarray}$$ (11)

These semi-empirical colors then define the semi-empirical pseudo-continuum, $\stackrel{\sim}{pc}_{\lambda} (T,\log g,\chi)$, at a given parameter set in the grid, following the method described in Paper I. In order to preserve the detailed information at the resolution of the synthetic spectra, a ``spectral function'', $\Gamma_{\lambda}(T,\log g,\chi)$, obtained by the ratio of the original model spectrum to the theoretical pseudo-continuum, is then multiplied with the semi-empirical pseudo continuum, in order to define the corrected spectra (see Appendix):

$$S^{\rm c}_{\lambda}(T,\log g,\chi) = \stackrel{\sim}{pc}_{\lambda} (T,\log g,\chi) * \Gamma_{\lambda}(T,\log g,\chi) \,.$$ (12)

This ``differential correction'' method should naturally preserve the original spectral features and the color differences of the models.

Identical tests as those performed in Sect. 4.2 for measuring the differential corrected colors indicate that, unfortunately, this ``differential correction'' algorithm fails to provide significant improvements over the conservation of differential colors attempted via the previous method. For models hotter than 3000 K, the residuals are still negligible (< 0.02 mag) and the two methods are really equivalent, but in the coolest dwarf regime, the new algorithm gives even worse results for UBV colors.

Thus, none of the two correction methods are able to preserve the differential color properties for the coolest star models to within the desired accuracy. Clearly, the definition and use of a pseudo-continuum is inadequate for such stars. Indeed, at these low temperatures, the presence of large and strong molecular absorption bands complicates the stellar spectra and hence also affects significantly the (broad-band) colors. Therefore, a pseudo-continuum defined from these colors as a smoothed (black-body) function cannot trace the flux distribution accurately enough. As an illustration, the theoretical spectra and their derived pseudo-continua are compared in Fig. 8 for two low values of the effective temperature: for $T_{\mathrm{eff}}$ = 3500 K (left panel), the pseudo-continuum (normalized in the K band) follows the flux variations quite accurately, whereas for 2200 K (right panel), the spectrum is too complex to be described by a continuous function such as a pseudo-continuum. Therefore, for temperatures less than $\sim$ 2500 K, a correction function defined from smoothed energy distributions is not suitable for color-calibrating theoretical spectra. In the future, a more reliable calibration and correction method for these complicated spectra of the coolest stars must obviously be based on the higher-resolution, more detailed observed flux distributions provided by eight-color narrow-band photometry (White & Wing 1978) or by spectrophotometric data (e.g. Kirkpatrick et al. 1991, 1993).

  

Figure 8: Comparison of theoretical spectra and their derived pseudo-continua for two values of $T_{\mathrm{eff}}$. The points indicate the monochromatic fluxes given by the theoretical colors