A corrected spectrum can therefore be simply computed by multiplying the original spectrum for a given set of stellar parameters (,, [M/H]) with the corresponding correction function at the same effective temperature. A scaling factor, , 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, , the magnitude mi in filter i is:
The color difference due to the variations in and are then given by
Since for the corrected spectra (LCB97),
Thus the differential colors are rigorously preserved by the correction procedure (i.e. ) if is constant between and . In practice, these rather severe constraints are well matched - and the corresponding are small - if 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 ( 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.
|Figure 4: Colors of original models of dwarfs compared to our empirical -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 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 for a ``dwarf'' model to 3.0 dex. Thus, all the spectra with lower than or equal to 2.5 and 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 -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 , 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, . For metal-content variations the results are presented in Fig. 6, where is plotted as a function of [M/H]. At low temperatures (< 2200 K) and low metallicities (< -2.0), differences as large as 2 mag are reached for U-B and B-V! For the other colors the residuals are smaller, but typical values of order 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 and (as indicated on the bottom-left panel). The color excesses are calculated as:|
|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|
These semi-empirical colors then define the semi-empirical
pseudo-continuum, , 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'',
, 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):
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 = 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 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 . The points indicate the monochromatic fluxes given by the theoretical colors|
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