4 Beat Cepheids

OGLE have also published data on SMC beat Cepheids. Because the knowledge of a (precise) second period adds a vital piece of information, these stars should be even more constraining than the single-mode Cepheids for extracting an M-L relation. In fact Kovács (2000) has used the two observed periods and $T_{\rm {ef f}}$ and radiative linear Cepheid models to infer luminosities and thus the distance modulus to the SMC.

Figure 6: SMC F/O1 stars: $\epsilon _1$ = $P_{1}({\rm calc}) / P_{1}({\rm obs})$ vs. the fundamental period for the SMC F/O1 stars. Left: distance modulus from choice a), Right: distance modulus from choice b).

In order to check the self consistency of the observational data and pulsation models we can make the following test on the SMC beat Cepheids. We take three of the four observed quantities, viz. $T_{\rm {ef f}}$, L, P_k and P_k+1 (k=0 for the F/O1 and k=1 for the O1/O2 beat Cepheids). From these three parameters (ignoring P_k+1 for the time being) we calculate the mass and then the second period $P_{k+1}({\rm calc})$. Then we compare this calculated period to the observed one ( $P_{k+1}({\rm obs})$) in Figs. 6 and 7. On the $\epsilon$ = $P_{k+1}(calc) / P_{k+1}({\rm obs})$ vs. P_k diagram, with the choices (A, B, C) of distance modulus and E(B-V) we observe the following facts:

Figure 7: SMC O1/O2 stars: $\epsilon _2$ = $P_{2}({\rm calc}) / P_{2}({\rm obs})$ vs. the first overtone period for the SMC O1/O2 stars. The upper pannels are for the choice a) of distance and reddening, whereas the lower pannels are for the choice b). Left: distance modulus from choice a) or b), Right: with $\delta {\rm Log}~L =-0.07$.

• For the F/O1 stars, the data fall along an almost horizontal line with little scatter. A similar result is obtained for the different choices, but $\epsilon$ is closer to 1 for choice (A). To get a self-consistent solution for these stars for choice (B), we have to increase the luminosity by $\delta {\rm Log}~ (L) = 0.12$. Decreasing the metal content (Z) to 0.001 also shifts the $\epsilon$ values in the right direction, but by itself it does not solve the discrepancy;

• For the O1/O2 beat Cepheids, the $\epsilon$ values scatter around a line with a slope of $\approx$0.03. There is only a limited range around $P_1 \approx$ 1.0 days, where self consistent solutions exist for the stellar parameters;

• The scatter on the $\epsilon$ vs. P_k plots are consistent with the observational noise in E(B-V), I and V. We note that with the help of surrogate data with the same noise as described in Sect. 2.3, we found that these error sources do not introduce systematic trends (like the slope of $\epsilon$).

For the second set of tests we allowed systematic shifts in Log L. For the O1/O2 Cepheids, the slope of $\epsilon$ strongly depends on the adopted $\delta {\rm Log}~L$. With $\delta {\rm Log}~L$ = -0.05 to -0.10, the slope is removed but the scatter of the points is increased, and $\epsilon < 1$ for all of the stars. Consistent solutions exist again only if the metallicity (Z) is decreased to 0.001. In the case of F/O1 stars the distance modulus has a less significant effect on the slope of $\epsilon$. The best agreement was found with $\delta {\rm Log}~L = 0.10$ which is opposite to the value we found for the O1/O2 Cepheids.

We have checked whether this discrepancy can be removed by allowing a wider range of initial assumptions on the input parameters. For our first set of tests the distance modulus was fixed, and we allowed a wide range in reddening ( $-0.1 < \Delta E(B-V) < 0.1$) as well as various changes in the composition and metallicity mixtures with the customized OPAL library. All these changes in the input data result in some vertical shifts in the $\epsilon$ vs. P_kdiagram, but not enough to get consistent solutions for the F/O1 stars. The metallicity would need to be decreased to Z=0.001 to get the mean value of $\epsilon$ to be 1. We also note that there is no significant difference in $\epsilon$ between the radiative and convective models.

Our conclusion agrees with the work of Buchler et al. (1996), but is in apparent disagreement with Kovács (2000). The reason for this apparent disagreement is that Kovács did not construct models with the observational parameters, but simply minimized what he called $\sigma$, viz. his measure of the deviation from observed to model periods, and in fact this sigma is not zero for many of his "solutions''. Furthermore in those cases where a solution can be found, the mass is determined with a very large uncertainty by the two period constraint, as already pointed out by Buchler et al. (1996).

Moreover, although this does not directly affect the absence of solutions, we remark that Kovács adopted reddening following that of U99. These reddenings are $\sim$0.01 larger than the mean reddening towards the SMC. This will marginally affect his temperature scale compared to ours. However the distance he derives is not in agreement with the distance adopted by U99 to the SMC, but is close to ours.

We note that the same trouble arises when we use the 3 observational data, (P_k, P_k+1, $T_{\rm {ef f}}$) and compute L and M. For many stars in the SMC sample there is no solution, i.e. no mass and luminosity can be found that satisfies these three observational constraints! The same difficulty appears when, instead, one tries to satisfy the 3 observational constraints ( P_k, P_k+1, L) to compute a $T_{\rm {ef f}}$ and M.

In the cases where there are solutions based on three pieces of observational data, they are generally not compatible with the fourth one, i.e. if the periods and $T_{\rm {ef f}}$ are given, the calculated luminosity and mass are not fully acceptable. Why there are no satisfactory solutions for the observed beat Cepheids in the SMC remains an unsolved puzzle that the introduction of turbulent convection in the linear codes has not resolved.