3. Surface brightness relations for Cepheid variables
Rather than assuming that Cepheid variables exhibit the same slopes on the various surface brightness - colour diagrams as the stable giants and supergiants, it is a better approach to determine the slopes from the Cepheid data themselves. This allows an independent check on whether or not Cepheids and stable giants and supergiants of similar effective temperatures do behave in an identical way on these diagrams, and it ensures that the surface brightness solutions (see Sect. 4) do not depend on a possible curvature of the surface brightness - colour relation at the very red end (M stars).
The surface brightness of a Cepheid in the V band, at a given phase, can be determined from its observed V magnitude and the displacement r from the mean radius according to the formula given by Thompson (1975):
The displacements can be calculated from the integrated radial velocity curve of the variable, and the mean value of the radius has to be known. Similarly, the surface brightness in the K band can be calculated from
Using photometric and radial velocity data and assuming a value for the mean radius, the variation of or with a colour index can thus be established for the variable, and the slope on this diagram be determined by least-squares fits. From this, the slopes on the or vs. colour diagrams are obtained by multiplication by -0.1, according to the definition of these surface brightness parameters:
3.1. The slope on the vs. diagram
The values of the slopes on this diagram have been established for 52 galactic classical Cepheids by Gieren (1988), using Thompson's method and the extensive and accurate photometric and radial velocity data of Gieren (1981a, b) and Coulson & Caldwell (1985). From these results, Gieren found that the slope m could be represented by:
where P is the pulsation period in days. However, the period dependence turned out to be only marginally significant. Moffett & Barnes (1987) conducted a similar study on a sample of northern galactic Cepheids and found somewhat more positive values for the slopes, but with no significant dependence on period either. While in the GBM study on the distances of galactic Cepheid variables a mean value from both determinations was adopted, we prefer here adopting the determination of Gieren (1988), because of the clearly higher accuracy of the radial velocity data he used in his work, which permitted to establish the radial displacements, and thus the surface brightnesses, with a better accuracy than in the work of Moffett & Barnes (1987). Neglecting the marginal period dependence, we adopt a mean value of for the slope shown by Cepheid variables on the vs. diagram, which is in excellent agreement with the value determined from the stable giants and supergiants in the preceding Section.
3.2. The slope on the vs. diagram
To determine the slope shown by Cepheid variables on this diagram, we chose 10 Cepheids with very accurate K band observations by Laney & Stobie (1992), spanning a period range between 4 and 39 days, which have also very accurate V band light curves and radial velocity curves by Coulson & Caldwell (1985), Gieren (1981a, b), and by Gieren et al. (1996). These stars are listed in Table 2.
Table 2. Slopes of near-infrared surface brightness relations from Cepheids
Here, a potential problem may arise, due to the use of different infrared systems, namely Carter's one, on which Laney & Stobie measurements are defined, and Johnson's one, on which stars with measured angular diameters are measured. As Bessell & Brett (1988) do not give definitive conversion relations between these two systems (they compared the Johnson system to an older SAAO system, defined by Glass), we have conducted a comparison on 87 common standards. Results are the following:
with and .
with and .
As the slope of the first relation and the intercept of the second one are not significant, we finally adopt:
with and .
with and .
We conclude that the K filters (including atmosphere and detectors) are almost identical (the same conclusion was reached by Bessell & Brett), while the J filters slightly differ. We will test in Sect. 3 whether this has a measurable effect on the adopted surface brightness relations.
Since the V and K observations were not obtained simultaneously, we adopted the procedure to fit Fourier series of appropriate orders to the K light curves and from this calculate the K values at the phases of the observed V magnitudes. The precision of such a K magnitude is generally better than 0.01 mag. Also, there was little problem with the adopted period values because these are well established and the epoch difference between the Laney & Stobie K observations and the Gieren and Coulson & Caldwell V and radial velocity observations is quite small. From this, we established and pairs at different phases and calculated the slope of the relation followed by each variable. The resulting values are given in Table 2. For all the variables, we found excellent linear relationships with typical correlation coefficients of 0.998. A typical example is given in Fig. 1. These excellent linear relationships indicate that the adopted mean radii, which were taken from Laney & Stobie (1995), are very nearly correct, although the adopted value of the mean radius is not critical in the calculations of the surface brightness: for the longest period Cepheids, where the relative radius variation is largest, a change of the adopted radius of 15% produces a change in the slope of 0.003. A plot of the resulting slope values against pulsation period shows that there is no significant dependence of the slope on period, and we adopt a constant mean value of for the slope shown by Cepheids on the vs. diagram, which is valid at least for the period range from 4 to 40 days. Within , this value is compatible with the one found to hold for giants and supergiants in the preceding Section.
3.3. The slope on the vs. diagram
To determine the slope shown by the Cepheids on this diagram, we used the same variables given in Table 2. We took the values from the tabulation of Laney & Stobie (1992) and calculated the surface brightnesses in the K band from the K magnitudes and the displacements calculated from the radial velocity curves at the phases of the J and K observations. For all the variables, a linear relationship between and is followed during the pulsation cycles, but there is more scatter than in the near-dispersionless relationships between and , which reflects itself in a larger scatter among the slope values determined for the different variables, which are given in Table 2. However, as in the case of the other diagrams discussed above, there is no significant dependence of the slope on pulsation period. From our determinations, we adopt as the appropriate value for Cepheid variables for the slope on the vs. diagram, which again is in excellent agreement with the value determined in the preceding Section from the stable giants and supergiants.
3.4. Adopted surface brightness relations for Cepheid variables
We now adopt the slopes derived from the Cepheid variables themselves in the preceding paragraphs and force these slope values separately to the same samples of giants and supergiants which were used to establish the surface brightness relations of Sect. 2. We do not apply the conversion relations between Johnson and Carter infrared photometric systems, but test the influence of such conversions. All this yields the zero points and standard deviations given in Table 3. Note that the supergiants zero point of the vs. relation is obtained after rejection of discrepant data for HR 7735 (without this rejection, the zero point is 3.924 and the rms 0.043), while the giants zero point of the vs. relation is obtained after rejection of HR 6056 and HR 6705 (without these rejections, the zero point is 3.953 and the rms 0.010).
Table 3. Mean zero points of the 3 surface brightness relations
Clearly, there is no significant zero point difference between the giants and supergiants in any of these relations. As our final value , we adopt a weighted mean of the six zero point determinations, which yields . Should we correct Johnson's values to Carter system, the zero points in the last line of Table 3 would become 3.951 and 3.947, leading to an unchanged weighted mean of the six zero points.
This new value 3.947 is slightly smaller than the GBM adopted value of , which was an average of values coming from model atmosphere calculations (), and from an effective temperature scale established from the work of Pel (1978) on short-period Cepheids (). Using the Pel data on 17 Cepheids with well determined mean colours, together with the mean slope of -0.380 and the reddening law adopted in this paper, the modified zero point of the vs. relation turns out to be , in excellent agreement with our adopted surface brightness zero point from the measured angular diameters of giants and supergiants. We therefore conclude that the value from the model atmosphere calculations is too high.
Another possible check on our surface brightness zero point comes from the angular diameter of the Cepheid Gem which was measured with the lunar occultation technique by Ridgway et al. (1982). This is so far the only Cepheid for which a direct angular diameter measurement has been obtained. From the measured value of milliarcsec and the magnitudes of the variable corresponding to the pulsation phase at the time of the angular diameter measurement as listed by Ridgway et al., together with a colour excess of (Fernie 1990), we find from Eqs. 1 and 2 and our adopted slopes a zero point of from the vs. relation, a value of from the vs. relation, and a value of from the vs. relation, respectively. The quoted uncertainties correspond to the uncertainty of the angular diameter measurement (which is, unfortunately, rather large). Clearly, within these uncertainties the zero point values agree with the zero point derived in our present study and lend some further support to our hypothesis that Cepheids and nonvariable giants and supergiants of similar colours do follow the same surface brightness - colour relations.
The final surface brightness relations we adopt for Cepheid variables are then the following:
These relations are shown in the three following figures, one for each relation, namely Figs. 2 - 4. Superimposed are the data points for the 28 stars with measured angular diameters with different symbols described in each figure caption.
3.5. Validity of the adopted reddening law
From the slope ratios, we can obtain a check on the adopted reddening ratios. Indeed, (resp. ) should be the same using any of the relations. We get:
Using our adopted value of , this gives:
The first of these results justifies our choice of from Hindsley & Bell (1989), among various discrepant values discussed in Sect. 2.3 of this paper. The second result is slightly larger than our adopted value 0.17, but other choices would have led to even smaller values of this ratio. Given the relatively large uncertainty of the value derived from the slope ratio in this case (a ratio of small numbers is involved), we do not regard this slight discrepancy as real.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998