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Astron. Astrophys. 354, 77-85 (2000)

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4. Results

4.1. Dependence of the [FORMULA] term on photometric bands

The main aim of the present analysis is to provide tight constraints on the [FORMULA] term adopted in the CORS method. This term quantifies the area of the loop performed by the variable in the [FORMULA]-color plane and therefore provides an estimate of the failure of the BW assumption that phases of equal color are also phases of equal temperature. In fact, the area of the loop described by the Cepheid in this plane supplies fundamental information on the variation of both effective gravity and effective temperature values along the pulsation cycle (Caccin et al. 1981).

Fig. 2 shows the [FORMULA] values we obtained by adopting the selected magnitude and color combinations (see labels) for both static (left panels) and effective (right panels) models. The first interesting outcome is that [FORMULA] values attained by static models are systematically smaller than the values of the effective models. This can be easily explained by the fact that the area of the color-color loops performed by effective models is larger than that of the static ones. This difference is caused by the sudden changes in the acceleration term (see Sect. 2) during the phases of rapid expansion and/or contraction. This larger excursion implies not only a difference in the area of the loop but also a change of its shape.

[FIGURE] Fig. 2. [FORMULA] values as a function of the logarithmic period for the whole sample of Cepheid models. The left panels show the models transformed into the observational plane by adopting the static gravity, while the right ones the models transformed by adopting the effective gravity.

The [FORMULA] values of effective models present substantial differences between different color pairs, and indeed the values attained for [FORMULA] colors are at least a factor of three smaller than the values for [FORMULA] colors. However, we note that radius evaluations based on colors which present large [FORMULA] values are not a priori more reliable than the evaluations based on colors with small [FORMULA] values. In fact, in the following we show that radii based on [FORMULA] colors are more in agreement with theoretical radii than the radii based on [FORMULA] colors. At the same time, large [FORMULA] values do not a priori imply that the CORS method is a major breakthrough in radius evaluations when compared to the BW method. In fact, radius estimates based on the BW method in [FORMULA] and on the CORS method in [FORMULA] are in very good agreement with theoretical radii, and the discrepancy for both of them is smaller than 10%.

4.2. Dependence of radius estimates on photometric bands

The radius estimates of effective models in different photometric bands are plotted in Fig. 3. Left and right panels show the radii evaluated by neglecting (pure BW method) and by including (revised CORS method) the [FORMULA] term respectively. To evaluate the accuracy of radius estimates based on different methods, in this figure we plotted the ratio between "computed" and "theoretical" radii. Data plotted in the left panels show quite clearly that BW estimates based on [FORMULA] and [FORMULA] bands are in very good agreement with theoretical radii, and indeed the discrepancy is systematically smaller than 10%. On the other hand, the radius evaluations in [FORMULA] are systematically smaller than the predicted ones and the discrepancy is of the order of 30% close to [FORMULA]. Thus confirming the empirical evidence originally pointed out by Welch (1994) and by Laney & Stobie (1995) that the use of [FORMULA] colors or infrared bands ensures more accurate measurements of Cepheid radii.

[FIGURE] Fig. 3. Ratio between "computed" and "theoretical" radii as a function of the logarithmic period. The left panels display the radius evaluations based on a pure BW method, while the right ones the radius evaluations based on the revised CORS method. The radius estimates plotted in this figure refer to models transformed by adopting [FORMULA].

This result is further strengthened by radius estimates obtained by means of the revised CORS method (left panels). In fact, the discrepancy between "computed" and "theoretical" radii is generally smaller than 10% when both [FORMULA] and [FORMULA] bands are adopted. At the same time, it is worth noting that radius determinations based on optical bands -[FORMULA]- present a discrepancy smaller or equal to 20% over the entire period range. The results of our numerical experiments suggest that by adopting NIR bands the radii evaluated through the revised CORS method present on average the same accuracy of the radii based on the pure BW method. However, the former method supplies more accurate radius determinations than the latter one when optical bands are adopted. Thus supporting the plausibility of physical and numerical assumptions adopted in the revised CORS method.

We will now focus our attention on the choice of the photometric bands which should be adopted for providing accurate radius determinations. Fig. 4 shows the comparison between "computed" and "theoretical" radii in the [FORMULA] plane. Data plotted in the top and in the bottom panels display radius estimates based on the pure BW and on the revised CORS method respectively. The main outcomes of this comparison are the following: 1) the slope of the PR relation, as already noted by Laney & Stobie (1995), becomes steeper when moving from optical to NIR bands. 2) Radius estimates based on NIR/optical bands overestimate/underestimate theoretical radii. These results apply to radius evaluations based on the pure BW method and on the revised CORS method, thus supporting the evidence that this "photometric drift" is not an artifact of the method adopted for estimating the radius. At the same time, data in the bottom panel of Fig. 4 suggest that radii obtained by averaging the estimates in the [FORMULA] and in the [FORMULA] bands are much less affected by systematic errors than the radius evaluations only based on NIR bands or on optical bands.

[FIGURE] Fig. 4. "Computed" and "theoretical" Cepheid radii as a function of the pulsation period in a [FORMULA] plane. The top panel shows the radius estimates obtained by adopting a pure BW method, while the radii plotted in the bottom panel by adopting the revised CORS method. Radius estimates based on different magnitudes and/or colors are displayed with different symbols.

4.3. A theoretical estimate of the [FORMULA] term

Even though previous results supply useful suggestions for improving the accuracy of radius measurements, the collection of both NIR and optical data for a large Cepheid sample is not a trivial observational effort. As a consequence, we decided to improve the approach suggested by RBMR for evaluating the [FORMULA] term. Since [FORMULA] is the area of the loop performed by each variable in the [FORMULA]-color plane, the idea is to compute [FORMULA] along the pulsation cycle directly from observations. However, the surface brightness depends on both [FORMULA] and [FORMULA], and therefore two relations should be inverted for deriving [FORMULA]:


where [FORMULA] and [FORMULA] are two arbitrary colors. Unfortunately this problem does not admit a general solution over the whole parameter space, since the same color can be obtained for different pairs of [FORMULA] and [FORMULA] values. This notwithstanding, it is still possible to find a local solution. Fig. 5 shows the surface covered by synthetic models in the 3D spaces [FORMULA] and [FORMULA] respectively. Data plotted in these figures show quite clearly that theoretical models populate a well-defined region of the quoted space. Therefore by performing a [FORMULA] degree polynomial fit to the data it is possible to invert the two relations governing the [FORMULA] and the [FORMULA] colors as a function of temperature and gravity. The results of the polynomial approximations are presented in the appendix.

[FIGURE] Fig. 5. Top plot: surface covered by the sample of theoretical models adopted in this investigation in the 3D space [FORMULA]. In order to make clear the dependence of [FORMULA] colors on both temperature and gravity the loop performed by each variable in the [FORMULA] plane (left panel) and in the [FORMULA] plane (right panel) are also plotted. Bottom plot: same as above, but in the 3D space [FORMULA].

On the basis of these relations we can estimate the surface brightness [FORMULA] directly from the following equation:


where the symbols have their usual meaning. The constant term depends on the photometric system used, but in our application it is not relevant, since the surface brightness in Eq. (6) appears as a derivative.

By taking into account this new theoretical calibration we applied once again the CORS method to the sample of synthetic models for estimating the Cepheid radii. Fig. 6 and 7 show the results of these calculations. Data plotted in these figures support the evidence that:

1) the theoretical calibration of the surface brightness we developed is intrinsically correct. In fact, the discrepancy between "computed" and "theoretical" Cepheid radii is systematically smaller than 7%. The only exception to this behavior is the radius of the model at 7[FORMULA] and [FORMULA] K which shows a well-defined bump along the rising branch. This evidence suggests that CORS estimates of Bump Cepheid radii could be affected by systematic errors. However, data plotted in Fig. 1 show that the outermost layers of this model undergo sudden gravity changes close to the bump phases. During these pulsation phases the assumption of hydrostatic equilibrium is no longer valid and therefore both the bolometric corrections and the colors obtained by adopting static atmosphere models should be regarded as suitable average estimates of the actual properties. It can be easily shown (Bono 1994) that this limit is mainly due to the effective gravity, since this quantity is estimated by assuming both radiative and hydrostatic equilibrium. The effective temperature only depends on the assumption of radiative equilibrium but the departures from the radiative equilibrium are, under the typical conditions of a pulsation cycle, quite small. These leading physical arguments suggest that the bump phases should be neglected in the CORS analysis.

2) In comparison with "theoretical" radii the "computed" radii do not show any systematic shift. This result suggests that the [FORMULA] terms evaluated by adopting the theoretical calibration -based on the polynomial approximations of [FORMULA] and [FORMULA] in the color-color plane [FORMULA] and of the BC in the [[FORMULA]] plane- are more accurate than the [FORMULA] terms obtained by means of the empirical calibration.

3) Accurate radius determinations can be obtained by adopting photometric data in three optical bands.

[FIGURE] Fig. 6. Top panel: [FORMULA] terms as a function of the logarithmic period obtained by adopting the revised CORS method and the theoretical calibration of the surface brightness. Bottom panel: similar to top panel but refers to the ratio between "computed" and "theoretical" radii.

[FIGURE] Fig. 7. Period-Radius relation in a log-log plane. The radius estimates have been obtained by adopting the revised CORS method and the theoretical calibration of the surface brightness. See text for further details.

Finally, we mention that following a referee's suggestion we applied the revised CORS method to theoretical curves which mimic real observations. In particular, we performed several numerical experiments by sampling the theoretical [FORMULA], [FORMULA], [FORMULA], and radial velocity curves with 20-30 phase points randomly distributed along the pulsation cycle. To account for observational uncertainties the points were spread out by assuming typical photometric and spectroscopic errors and then fitted with up to 7 Fourier terms. Interestingly enough, we find that radius estimates based on these curves still present a discrepancy [FORMULA] 7% when compared with theoretical radii. Therefore this uncertainty can be assumed as a plausible upper limit to the intrinsic accuracy of radius determinations based on the revised CORS method.

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Online publication: January 31, 2000