3. Test of the revised CORS method
In the following we briefly summarize the main features of the CORS method. The reader interested in a comprehensive discussion on the adopted physical and numerical assumptions is referred to RBMR and references therein. The CORS method relies on the definition of surface brightness :
The solution is found by differentiating Eq. (2) with respect to the phase, then by multiplying the result for a color index, e.g. , and eventually by integrating along the full cycle.
where is the phase, P is the pulsation period, is the radius (in solar units) at a given phase , u is the radial velocity, p is the radial velocity projection factor (Parsons 1972; Gieren et al. 1989; Sabbey et al. 1995) and a is a constant equal to .
The numerical solution of Eq. (4) supplies the unknown quantity . In order to evaluate the radius as a function of the phase we adopt Eq. (3) and finally the mean radius is estimated by averaging along the radius curve. By neglecting the term in Eq. (4), we obtain the pure Baade-Wesselink method which requires a radial velocity, a magnitude and a color curve for each individual variable. A more precise radius determination can be obtained by including the term. In fact, Sollazzo et al. (1981) and RBMR demonstrated that the inclusion of this term improves the accuracy of radius estimates, provided that is evaluated at each pulsation phase. The term was included in the original CORS method (Sollazzo et al. 1981) by adopting the empirical photometric calibration of the Walraven system provided by Pel (1978), and in the revised CORS method (see Sect. 2.3 in RBMR) by adopting the empirical calibration of the reduced surface brightness () as a function of , provided by Barnes & Evans (1976). This change allowed RBMR to apply the CORS method to a large sample of Cepheids for which photometric data in the conventional BVRI bands were available. It is worth underlining that both the original and the revised CORS method do require two color curves but the latter method, thanks to the new calibration, can supply radius estimates of Cepheids for which are available two different pairs of Johnson/Cousins color indices.
In order to test the accuracy of the term evaluation we apply the two previous approaches to synthetic light, color, and radial velocity curves. In particular, we adopted theoretical periods and the synthetic curves, covered with 125 points, were fitted with Fourier series which include up to 31 terms (15 sine, 15 cosine plus a constant term) and eventually the quantities B and were evaluated as well. We adopted a large number of both points and Fourier terms, since we are interested in testing the accuracy of the revised CORS method by adopting theoretical templates which should not be affected, within the intrinsic uncertainties, by systematic and/or deceptive errors.
Since the modified CORS method requires an empirical estimation of the surface brightness as a function of a color, in this investigation we applied the calibrations provided by Fouqué & Gieren (1997) on the basis of stellar angular diameter measurements collected by Di Benedetto (1993) and Dyck et al. (1996), i.e.:
where F is the reduced surface brightness and is the Johnson R band. However, it is worth noting that the R photometric bandpass adopted by Castelli et al. (1997a,b) is the Cousins band. Therefore to account for the color difference between and the slope in Eq. (7) has to be replaced with 0.521 according to the transformation provided by Bessel (1979). The uncertainty on this color transformation, due to a twofold fortunate circumstance, has negligible effects on radius estimates. In fact, as discussed by RBMR, only the slope of the versus color calibration is taken into account in the revised CORS method, since the zero point does not affect the derivatives. On the other hand, a change of the order of 30% in the slope of Eq. (7), due to the additive nature of the term, causes a change of only 4% in the radius estimates.
A similar calibration - versus - was originally suggested by Di Benedetto (1995). However, for applying the revised CORS method to different colors, we adopted the multiband calibrations provided by Fouqué & Gieren (1997). We emphasize once again that the modified CORS method adopts one magnitude and two color curves (cases 2, 3, 5 below), whereas the pure BW method adopts one magnitude and one color curve (cases 1, 4, 6 below). On the basis of the selected bands we investigated the following combinations of magnitudes and colors:
These bands were selected because they are quite common in the current literature, and also because they give a proper coverage of both optical and NIR wavelenghts.
© European Southern Observatory (ESO) 2000
Online publication: January 31, 2000