5. Comparison with Freedman's solution
Equation (4) differs fundamentally from the Freedman procedure because all phase shift information is contained in the shape of the C correction function. The phase shift in any particular Cepheid is permitted to change as it will over the V phase. Whatever its average run for "normal" Cepheids is reflected in the C function as we have derived it.
Freedman's solution is
where denotes the phase shifted by +0.07. We have made numerical experiments using both Eqs. (4) and (5) to assess the accuracy of each method. We have carried out Eq. (5) on the six Cepheids in Table 1 at each of the 20 phases, separated by 0.05 phase units, so as to construct correction curves as a function of phase for Eq. (5) as we did for Eqs. (1) to (3), shown as Fig. 1. We found that the Freedman squeeze and phase shift method is excellent, except during the phase interval of the rising branch. This was to be expected because of the way the non-isomorphism of the phase shifted curve manifests itself, as explained in x 1. However, the phase interval is small in which the large deviations occur in the Freedman solution. The problem typically is between phase 0.85 to 1.00. The deviations can produce an error as large as 0.3 mag in this phase interval. This part of the light curve must be avoided when using the Freedman method which does not take into account the color variation with phase.
On the other hand, our method also has a moderately large scatter of the individual points in Fig. 1 about the adopted mean correction curve 1, amounting to an rms of 0.05 mag for a single measurement for the normal Cepheid curves in Table 1. Hence, if four random points are selected, and their resulting individual values determined using Eq. (4), the resulting mean found by averaging the four determinations will have an rms of 0.03 mag. The Freedman solution has similar errors if the bad phase interval is avoided.
To assess the accuracy directly we have selected 8 to 10 random phases for the four normal Cepheids in Table 1 and have applied both Eqs. (4) and (5) to the data to determine the individual values at each phase. The result is shown in Fig. 2. Plotted is the difference between the real as known for each Cepheid in Table 1 using the complete I light curves in the literature, and the calculated from Eqs. (4) and (5) at each random phase. Except for the bad phase interval mentioned above for the Freedman method, the scatter in both methods is similar. Table 3 gives the difference in and its rms value based on these numerical experiments using the 8 to 10 phase points for each star. In the actual HST programs we have had only 2 to 5 individual data points in I, hence the numbers in Table 3 should be increased by about a factor of 1.5 for the HST estimate.
The conclusion is that both methods are competitive, and that each is capable of delivering the important data with as few as four random epochs in I at an accuracy level of 0.03 mag due to this problem alone. However, in contrast to Freedman's approach the method presented here makes explicit allowance for the color change with phase as observed in Cepheid light curves.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998