4. The distance and radius of the cluster Cepheid U Sgr
In order to carry out a first test of the new surface brightness relations obtained in the preceding Section, we apply the Barnes-Evans optical and near-infrared technique to the cluster Cepheid U Sgr. This particular Cepheid has not only excellent observational data for this kind of analysis, but is furthermore a member in the open cluster M 25 and thus has an independent determination of its distance from cluster ZAMS-fitting which can be compared to the surface brightness results. U Sgr has also been used by Welch (1994) to test his calibration of the K, relation obtained from giant and supergiant angular diameters then available for calibration.
4.1. Observational data
Since the calibrating angular diameters are good to a few percent, we need observations of the highest possible quality to ensure that the resulting radius and distance are not degraded further. We use the V and observations of Moffett & Barnes (1984) on the Johnson system and combine them with the observations of Gieren (1981a) to produce a high-quality V light curve and colour curve of the Cepheid, consisting of 81 points providing excellent phase coverage. The Gieren R, I observations were obtained on the Cousins system and were transformed to the Johnson system using the relation (Cousins 1981):
In fact, the original Cousins relation has a mean offset of 0.03. But this offset varies e.g. with right ascension, and we found that for our particular star the appropriate value was 0.055, giving an excellent fit to the Moffett & Barnes curve. No zero point shift was necessary in V. We further use the excellent J and K light curves of Laney & Stobie (1992; 30 points) which were measured on the Carter system. Finally, we adopt the excellent CORAVEL radial velocity curve of the variable of Mermilliod, Mayor & Burki (1987). While the radial velocities and the optical observations were obtained nearly simultaneously, the near-infrared photometry was obtained a few years later, but in view of the apparently very stable and precisely known period of U Sgr, confirmed by the adoption by Laney & Stobie of Pel's period (1976), this does not introduce the possibility of a significant phase mismatch between the infrared light curves and the optical light curve and radial velocity curve (anyway, a phase mismatch, if existent, can be detected in the surface brightness solutions, representing one of the strengths of the Barnes-Evans technique).
4.2. The method
Using our new calibrations of the surface brightness parameter in V and K with colour (Eqs. 26- 28) and combining them with Eqs. 1 and 2, we get the following equations for the angular diameter of a Cepheid in terms of the different magnitudes and colours:
Note that while the adopted zero point 3.947 depends on the precise value of the adopted constant 4.2207 in Eq. 3, the resulting diameters are independent of it, therefore do not depend of a given choice of the bolometric absolute magnitude of the Sun.
The unreddened magnitudes and colours are calculated with the relations adopted in Sect. 2.3, using (Laney & Stobie 1992). We then get the angular diameter variation of U Sgr from each of these formulae from the pairs of a magnitude and colour index obtained at the same phases. While the V, and J, K pairs were obtained simultaneously, this is not the case of the V and K observations. In order to apply Eq. 35, we therefore fitted a Fourier series of order 3 to the Laney & Stobie K light curve, which gave an excellent representation of the measured data, and calculated from this the K magnitudes at the phases of the actually observed V datum points; from this, the values at the phases of the V observations were obtained. To illustrate the precision of this process, we show the resulting colour curve of U Sgr in Fig. 5. To obtain the linear displacements of the stellar surface at the phases of the photometric observations, we integrated the radial velocity curve adopting a projection factor of 1.365 which was calculated from the formula given by GBM. The adopted period value for U Sgr is 6.744925 days (Pel 1976). The mean radius and distance of the Cepheid variable is then obtained from a linear regression of the relation:
where is the mean linear diameter in AU, the displacement from the mean in AU, d the distance of the star in pc, and the angular diameter in milliarcsec.
The radius and distance values for U Sgr obtained from the three different surface brightness relations, and their errors, are summarized in Table 4. The corresponding plots for the three cases, showing a) the angular diameters together with the linear displacement curve and b) the linear displacements vs. the angular diameters (the slope of which gives the distance of the star) are shown in Figs. 6 - 11. There is no need to apply any phase shift between the linear displacement and the angular diameter curves, in none of the three solutions. It is impressing to see how the angular diameters obtained from the V, pairs adjust to the linear displacement curve obtained from the integrated radial velocity curve, in a diagram which is almost scatter-free. The error in the distance and radius is less than 1% in this case. This very small error is due to a combination of two effects: first, to the shallow slope of the vs. surface brightness relation, as compared to the vs. relation, which permits a more accurate correction for the temperature-induced variation of the surface brightness of a Cepheid, and second, to the much reduced sensitivity of the colour index to variations in gravity and microturbulence during the pulsation cycle, as shown by Laney & Stobie (1995). The situation is somewhat less favourable in the K, solution where the error is 3%, but this is still very good taking into account that only 30 J, K data points have been available for this analysis, and that the fit of the K light curve for the V, solution artificially reduces the observational scatter; with a comparable number of data as in the V, solution, the error would probably decrease to 1-2%. Also very importantly, there are no systematic effects in these two solutions employing a near-infrared colour index which could produce a significant difference in the radius and distance if one restricts the solution to a particular phase range, like descending or ascending branches of the light curve, which is generally not true for the V, solution. The results given in Table 4 are derived from bisector solutions to Eq. 37 which assumes equal errors in the angular diameters and linear displacements. In the case of the near-infrared solutions, this assumption is not critical, and simple least-squares fits produce the same radius and distance results (the difference is smaller than ). In the case of the V, solution, however, the difference between the bisector and least-squares solution is in the case of U Sgr and thus clearly significant.
Table 4. Surface brightness solutions for the distance and radius of U Sgr
There is clearly much more scatter in the optical V, solution. However, and perhaps surprisingly, the distance and radius obtained from the optical version of the method agrees very well with the results from the near-infrared solutions, albeit with an error about 5-10 times larger. Whether this is a coincidence or a general feature remains to be further investigated.
It is very satisfactory to see that the three different surface brightness solutions all agree within . A weighted mean for the distance and radius of U Sgr from the present results is pc (corresponding to a distance modulus of 8.87) and solar radii. The mean radius compares very well with the Laney & Stobie (1995) infrared value solar radii. The distance may be compared to the ZAMS-fitting distance modulus of M 25 which is 8.95 (Pel 1985). It may also be compared to the distance of pc obtained by Welch (1994) from his calibration of the K, surface brightness relation: we attribute the difference with our result to our improved calibration which was able to take advantage of an increased number of stars with accurate angular diameters, as well as to the inclusion of the Gieren (1981a) V data in the surface brightness analysis of this paper.
We will give a detailed analysis of possible systematic errors in the method in a follow-up paper (Gieren, Fouqué, & Gómez 1997), where a large sample of cluster Cepheids will be analysed in the same way, which will permit to identify possible trends and problems more safely than from the analysis of just one star. This discussion will include the problem of a possibly variable p -factor (Sabbey et al. 1995; Sasselov & Karovska 1994) which is common to all Baade-Wesselink - type methods. We just note here that the close agreement in shape between the linear displacement curve of U Sgr (derived in our study under the assumption that const) and the angular diameter curve of the star obtained from infrared photometry (see Figs. 8 and 10) lends support to the idea that const is a good assumption in the case of U Sgr: neglect of a significant variation of p during the Cepheid's pulsation cycle should have distorted the linear displacement curve from its true shape, an effect for which there is no indication in Figs. 8 and 10, and it thus appears that a variable p -factor is an effect of second-order importance in the determination of the radius and distance of U Sgr from the Baade-Wesselink infrared technique.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998