## 2. Derivation of veilingIn this section, we consider a spectral range small enough for the
excess and the extinction to be constant. The reference spectrum
## 2.1. The formalismIn the absence of noise, by definition where is the wavelength,
a scaling factor, where the index where is a vector whose components are and , i.e. the increment to vector whose components are and ; is the transpose matrix of the derivatives of with respect to the s, the upper bar over standing for the expected value. The element of matrix is written as: For the calculations, can conveniently be approximated by . At each step, the matrix and the vector are then recomputed. The procedure is very robust and it generally converges after a few iterations, if the initial values of the s are close to the correct ones. At the end of the iterative process, the variance of the s are the diagonal elements of the matrix . The reduced chi-square value is simply given by and should be close to 1 for a statistically correct fit. ## 2.2. Error analysisThe detailed error calculation is given in Appendix A. Generally, the reference star is much brighter than the object source, and its contribution to the noise can be neglected. Indeed, it is important to have a set of good quality reference stars which can be used in any veiling study. Under this assumption, for both photon and constant additive limited noises, within a very good approximation, the standard deviation (error hereafter) of the veiling is given by: and the relative error on the scaling factor can be written as: where is the signal to noise ratio on the object total spectrum flux and is the variance of the normalized reference spectrum expected value . measures the square of the spectrum contrast and is defined by: Clearly, for veiling values greater than 1, the veiling error increases as the square of the veiling which illustrates the difficulty to estimate large veiling values. These results are discussed in Sect. 2.4. The relative veiling error can also be written as: This formula can be useful to estimate rapidly the veiling error from the data and allows us to introduce the contrast spectrum in noise units (resp. ), which is a central quantity in bias studies. ## 2.3. Bias problemsThere are two main sources of bias in veiling calculation. The first is due to a bad estimate of the noise ratio between the object and the reference. The second is due to mismatches between the object and the reference spectra. ## 2.3.1. Uncorrect noise estimateLet us first study the noise problem. We assume for simplicity constant additive noises for the object and for the reference. Eq. (2) can then be rewritten as: where . Clearly, Eq. (9) shows
that the estimated veiling depends
on the input noise ratio between the reference and the object. If this
noise ratio is not correctly estimated, the veiling will be biased. We
derive an analytical expression of this bias in Appendix B1. We find
that it is a function of the veiling where the brackets stand for the mean value. This bias has a very simple explanation. Underestimating the noise ratio is equivalent to underestimate the reference noise , or equivalently, to overestimate the object noise . Let us concentrate for example on the reference. If is underestimated, the algorithm will interpret part of the reference noise in terms of high frequency signal, it will "see" the reference absorption lines apparently deeper than they really are and will tend to overerestimate the object veiling. On the contrary, if is overestimated, the algorithm will interpret part of the true signal in terms of noise, it will "see" the reference aborption lines less deep than they really are and will then tend to underestimate the object veiling. It is not always easy to estimate the true noise ratio between the reference and the object. But if one of the two sources has a very good signal to noise ratio, usually the reference, then is large and , as shown in Eq. (10). It is then convenient to set in Eq. (2): the derived veiling is biased, but the bias is negligible. On the other hand, if is large, [see Eq. (11)]: in this case it is better to set in Eq. (2). If both and are small, the bias can be very important even if the input noise ratio is not correct by only a factor of 2 (as shown in Figs. B1a and B1b). In case of doubt, one can minimize the bias by underestimating or overestimating the noise ratio , depending if is larger or smaller than , respectively. To conclude this section, we generally recommend to filter the spectra before veiling calculation, but the choice of the working spectral resolution must be examined case by case. The spectra between 5180 Å and 5220 Å presented in Fig. 1 of Hartmann & Kenyon (1990) are typical examples. Indeed, we can see the presence of high frequency noise superposed on lower frequency structures. Filtering these data by a factor of a few will considerably reduce the noise without affecting significantly the spectrum contrast, which in turn will greatly decrease the possible bias due to a bad noise estimate at a negligible cost in terms of the veiling error. ## 2.3.2. Mismatches between the object and the reference spectraThe other source of bias, regards mismatches between the object and
the reference spectra, and it is treated in details in Appendix B2. By
mismatches, we mean spectral differences between the reference
spectrum and the true underlying object stellar spectrum In the case of non noisy data, the validity of Eq. (12) can be
checked by computing the extreme veiling values
and
obtained from
and
infinite, respectively. If these two
values coincide, then Eq. (12) is correct. Otherwise, we must add to
Eq. (12) another bias which depends on the input
value. The distance between
and
is a good indicator to evaluate the
importance of mismatches between ## 2.4. The effect of the spectral resolutionIn this section, we study the effect of the spectral resolution on the veiling value and the associated noise. For this purpose, we use a high quality spectrum of the K7V star HD 201092, obtained at the 1.93m telescope of the Observatoire de Haute Provence (France) with the instrument ELODIE, at (see Baranne et al. 1995) and for which and . Fig. 1 shows this spectrum between 4000 Å and 6800 Å, normalized to its local continuum at and then filtered at .
Defining as we did so far, the veiling as the ratio between the excess and the local stellar continuum, does not provide an absolute quantity, because the level of the local continuum is resolution dependent. To study the effect of the spectral resolution on the veiling value, we have selected a spectral band of 40 Å centered at 5200 Å from our K7V star, normalized to its local continuum to simulate the reference, and added a veiling of 1 to the reference to simulate the object. We then degrade the spectral resolution of both the object and the reference by gaussian filtering, and estimate the veiling, renormalizing first the reference spectrum to its new local continuum. The veiling is found to vary by about 10% when the spectral resolution varies from 40000 to 500. However, the product of the veiling with the spectrophotometrically calibrated local continuum of the reference spectrum, which measures the excess, will obviously remain unchanged. Defining the veiling as the ratio between the excess and the mean reference flux (see HHKHS) would result in a quantity little dependent on the spectral resolution. The veiling error, Eq. (5), is inversely proportional the product
which is sensitive to the spectral
resolution
© European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |