- 4.1 Correction procedure and conservation of the original differential properties
- 4.2 Correction of M dwarf model spectra
- 4.3 Shortcomings of the correction algorithm

(5) |

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
*m*_{i} in filter *i* is:

(6) |

(7) |

The color difference due to the variations in and are then given by

(8) |

Since for the corrected spectra (LCB97),

(9) |

(10) |

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.

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.

(11) |

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):

(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 = 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 |

Copyright The European Southern Observatory (ESO)