From the above definition it can be shown that
where 
is the equivalent width of a photospheric line in
the spectrum of a template star, i.e.
a star for which the veiling is zero and
is the equivalent width of the corresponding photospheric line
in the spectrum of the T Tauri
star.
The veiling can be determined by measuring the equivalent widths of a set of
photospheric lines in
the the spectrum of the T Tauri
star and in that of a template star of the
appropriate spectral type. However, this method works better when there
is a range of photospheric
lines with different excitation potentials [Basri & Batalha 1990],
which is not the case here. An alternative method
had therefore to be used. A number of methods have been discussed
in the literature. Basri & Bertout (1989) use an
auto-correlation method, Hartigan et al. (1989) use a
-algorithm to fit pixel by pixel the observed spectrum with that
of a template star plus a featureless continuum, Guenther & Hessman
(1993a) use a cross correlation method based on a technique
discussed by Tonry & Davis (1979), where the cross
correlation function between the T Tauri
star spectrum and that of a template
star is computed along with the auto correlation function of the spectrum
of the template star 
. In this method, the veiling is determined by
computing the ratio of
the peak of the auto correlation function (pac) to the peak of the cross
correlation function (pcc) and subtracting unity from it, ie.
Given the data set presented here, the cross correlation method was selected to compute the amount of veiling present in the T Tauri stars where a photospheric spectrum could be identified. A section of the spectrum containing only continuum and photospheric lines was chosen and cross correlated with the corresponding wavelength region of the template star.
The method described above was implemented in IDL.
As discussed by Tonry & Davis (1979) it is important to filter the data before computing the cross correlation function. The aim of filtering is to remove those features that are not relevant for the veiling analysis, that is to say removing low frequency spectral variations as well as high frequency components due to noise beyond the resolution. This is done by means of a band pass filter which for the Pa Beta spectra was built using the IDL built-in routine digital_filter, with cut frequencies 0.0625 and 0.25, in units of the Nyquist Frequency, corresponding to periods of 8 and 2 pixels respectively.
The uncertainty in the computation of the veiling results from the
uncertainties in the
determination of the peaks of the cross and auto correlation functions. These
uncertainties were estimated from the root mean square ( rms) of the
antisymmetric component
of these functions [Tonry & Davis
1979]. Given equation 4.3
and noting
that the auto correlation
function is symmetric, ie. 
,
where, 
, therefore,
In order to test the cross correlation method outlined above, spectra of
template stars were veiled by a known amount and different levels of
noise were added. The method was then
applied to check whether the correct amount of veiling was
recovered. Figure 4.2 shows the spectrum of the K7V template star
artificially veiled and with a decreased signal-to-noise ratio, as well
as the original, therefore not veiled, observed spectrum. In each of
the panels the signal-to-noise ratio decreases towards the bottom (by 2%, 5%
and 10% respectively). Spectra in the top panel
have a veiling of 0.5 and 1.5 is the veiling for the spectra in the
lower panel. By applying the cross correlation method to the wavelength
region between about 1.2845 
to about 1.2890 
the following results were
obtained: top panel top to bottom spectra 
, 
and 
; lower panel top to bottom spectra 
,
and 
. Comparing these results with the known
amount of veiling present one sees that there is a good agreement. As
expected, the
noisier the spectrum the larger the difference between the computed and the
real amount of veiling. The associated uncertainties also increase.
Figure 4.2: Top panel - Pa Beta spectrum of Gl380
as observed (r=0); veiled (r=0.5) and noisier (by 2%, 5% and 10%). Bottom
panel - Same as above for veiling r=1.5
All tests performed showed that this is a reliable method to compute the amount of veiling from the spectrum of a T Tauri star using the data set analyzed here.
For the Pa Beta
data, the wavelength region chosen to perform the cross
correlation analysis was from about 1.2845 
to 1.2890 
in order to
avoid the emission line in the centre of the observed spectral region
and so that three lines of photospheric origin are present.
Unfortunately the cross correlation method could not be applied to the determination of the veiling in the Br Gamma wavelength range. This is due to the fact that, at these wavelengths, only two photospheric lines could be used for the veiling computation, allied with the fact they are somewhat blended and that the signal-to-noise ratio in the spectra is, in general, lower than that obtained at Pa Beta wavelengths. Nevertheless, an estimate of the veiling was computed, even though the method used was a lot less precise. After deciding on the correct template star to use, its spectrum was veiled by a known amount and the result subtracted from the T Tauri star spectrum. This procedure was carried out starting from a veiling of zero up to an amount of veiling clearly not consistent with the data, in steps of 0.1. A range of values for the veiling consistent with the data was chosen, as judged by eye! The amount of veiling was taken to be the mean of the above values with an estimate for the error being the difference between the mean and the most extreme value.
