2. Veiling derivation: the state of the art
The veiling of a T Tauri star has been defined as the ratio of its excess flux to either the local photospheric continuum of the comparison template, named r, or to the mean flux of the template spectrum within the working band, named R (HHKHS). In the veiling derivation process, the template spectrum is assumed to represent the object underlying unveiled spectrum.
Two main classes of algorithms have been proposed to extract the veiling from CTTS spectra. The first class was introduced by HHKHS and is based on a point to point comparison between the photospheric absorption lines of the object and those of the template star through an appropriate minimization. The second class is based on a comparison between the lines energy content of the object and that of the template star. The comparison is made either through the concept of equivalent width or through the concept of cross and auto correlation (Basri & Bertout 1989, Basri & Bathala 1990, Guenther & Hessman 1993). In principle, these two algorithmic classes are equivalent from the point of view of the signal to noise ratio. Although the lines energy content method is much easier to use than the point to point method, we hardly see how this global approach can give an insight to the problems of bias inherent to any veiling derivation (see CH99 and below). For this reason, although the question deserves a more detailed study, we think that the point to point approach of HHKHS should, whenever possible, be preferred.
HHKHS proposed to extract the veiling r from a point to point comparison between photospheric absorption lines of the object spectrum and those of an appropriate template spectrum , each normalized to their local continua. In practice, the comparison is made over a spectral interval small enough (a few tens of Angstroms), where r can be assumed wavelength independent. For constant noise on both the object and the template spectra, the problem of veiling derivation consists in minimizing, as a function of the two unknowns p and r, the quantity:
where the index k stands for the sampled wavelength domain, is a window function, p is a scaling factor, is the inferred object spectrum noise and is the inferred noise ratio between the template and the object spectra.
Let us summarize the conclusions of CH99 with respect to the bias in veiling calculations from Eq. (1). There are two classes of bias coming from: 1) bad noise ratio estimates and, 2) spectral mismatches of any kind, real or due to systematic errors, between the object and the comparison template. Recently, Guenther et al. (1999) have shown that the presence of strong magnetic fields in T Tauri stars may lead to wrongly determined spectral types and may induce veiling overestimates. Let be the calculated veiling and be the difference between the true and the calculated veiling. Let us assume for the moment that there is no spectral mismatches. The veiling bias is obviously equal to zero if the input noise ratio in Eq. (1) represents the real noise ratio. On the other hand, the relative veiling bias is bounded by a positive upper limit and a negative lower limit:
where means in Eq. (1) and (resp. ) is the variance with wavelength - square of the spectral constrast - in noise units of the template (resp. object) spectrum in the selected spectral bandpass, i.e.:
The 's are the parameters scaling the bias. Eqs. (2) and (3) clearly show that when the object and the template noises tend to zero, the upper and the lower limits of the veiling bias also tend to zero and the solution of Eq. (1) becomes more and more independent of any input parameter . This simple result allows us to consider the problem of spectral mismatches.
Let us now assume relatively not noisy spectra ( and ). Any variation of the calculated veiling as a function of the input value in Eq. (1) must be attributed to spectral mismatches between the object and the template spectra. In this case, as we do not know the true object underlying photospheric spectrum, it is no longer possible to strictly bound the object veiling with minimum and maximum values. But, it is still possible to compute the "extreme" veiling values and . CH99 has shown that if the difference is equal to zero, then Eq. (1) is exact and the bias is dominated by the apparent veiling of the template, if any [see Eq. (12) of CH99]. Otherwise, the difference can be interpreted as the goodness of the fit to the veiling equation. In practice, there is a priori and to the knowledge of the author, no objective reasons to choose one solution rather than the other one ( or ) or any intermediate solution. However, we can always define an upper limit, necessarily somewhat arbitrary, to the relative difference between the extreme veiling values, , for example 10 to 20%, below which we consider that the template spectrum is a "good" representation of the object underlying photospheric spectrum and hence the solution is acceptable. Under these conditions, we propose to adopt the mean value between and as representative of the true veiling and their half difference as its associated error.
© European Southern Observatory (ESO) 1999
Online publication: September 13, 1999