3. Stability of the method
We want to study here the characteristics of the method, that is the sensitivity of the method to the parameters and data present in Eq. (13). First of all, the left side of Eq. (13) is a function of the following parameters: observational data (V, B-V, V-R, u); constants (c and p); the variable R. This function has a zero at a value which is determined numerically; it is important to study the behavior of such a function, with respect to R. Fig. 1 gives the plot of the above function for ; it is a well-behaved function whose zero can safely be determined. The plot varies of course according to the star, but the general shape is similar.
Since the terms B and in Eq. (12) are additive terms, their effect is to move the whole curve by a vertical shift, thus changing the value of . This explains the importance of the loops for both B, the area in the plane (V, B-V) and C, the area in the plane (B-V, V-R). However, the observational data enter in these quantities only globally, that is all data together combinate to give B (and C), and this means that errors on individual points have less influence on the final value of B and C, and hence of . However, the evaluation of B and C is very dependent on the regularity of the curve: if the noise of the data is high, and the data have a large scatter around the fitted curve in the planes (, V), (, B-V), (, V-R) then the area of B and C may be not well determined. Fig. 2 shows the plot of the magnitude-color and color-color loops for . From this figure it is clear that the scatter in color-color loop is more critic with respect to that of the color-magnitude loop, so that the term will be more influenced by the eventual poor quality of the data. In any case, since the area described by the color-color loop is very small (due to the characteristic of to be a correction term) it is reasonable to expect that the error introduced in this way will be very small; an occurrence already noticed by Sollazzo et al. (1981), (see their Fig. 8).
For what concerns the dependence on the projection factor p, Fig. 3 shows the value of obtained for , W Sgr and SZ Tau, for five different values of p: a change of 0.05 in p corresponds to a change of 3.5% in .
This latter test is particularly important, because there is now much doubt about the fact that p, the projection factor has actually a constant value. The old Parson's (1972) constant value p=1.31 has been used for over two decades essentially for lack of better knowledge, and Gieren et al. (1989) using models by Hindsley and Bell (1986) have already used a value of p variable with the period as we quoted in Sect. 2.3. However, it is likely that p also varies along the pulsational cycle for a given period, as already argued by Hindsley and Bell (1986) and more recently shown by Sasselov and Karovska, 1994 and Sabbey et al. (1995).
Following referee's suggestion, we have therefore extended our tests as follows: for a particular star, the same discussed by Sabbey et al. (1995), we have solved Eq. (13) using the variable projection factor p as a function of the phase, given in Fig. 10 in the paper by Sabbey et al. (1995). The result of our test is reported in Table 1 ; we found a percentage variation for the radius of about , i.e. a value equal to the one quoted by Sabbey et al (1995).
Table 1. Results of our test of using a variable projection factor p from Sabbey et al (1995) for the star .
Bearing in mind that this effect certainly exists and introduces further uncertainty in the Baade-Wesselink radius, we hope that in the future further and more detailed indications on the variation of p with respect to the phase will be available.
For the present work we choose to use a constant value for p (p=1.36) either because it is appropriate for CORAVEL radial velocities (Burki et al., 1982) and approximates very well the p depending from the period by Gieren et al (1989) for a large range of periods, or because in this way we can compare our results with those ones from previous works.
However the problem of the correct estimate of the projection factor p for any star is still open.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998