3 Discussion

3.1 Which choice for distances and reddening?

We recall that the main difference between (B) and (C) is the adoption of mean SMC and LMC reddenings in (B) and (OGLE) reddening maps in (C). As we have seen in Figs. 1-3 there is no significant improvement with the use of reddening maps, neither in the P-L residuals, nor in the scatter in the final M-L plots. Choices (B) and (C) are essentially equivalent except for a tiny systematic shift parallel to the distribution.

As seen in Fig. 1, with the observational constraints (A) there is a systematic shift to higher luminosities and higher temperatures for the observed stars. Taken at face value, choice (A) would indicate that evolutionary calculations are far from the representative which we deem unlikely given the general agreement of several independent calculations (cf. Sect. 4). The sensitivity of the M-Lrelations to metallicity would have to have been largely underestimated, too. On the other hand, with the "long'' distances to the clouds of choices (B) and (C), the situation is much more satisfactory, as we shall see.

In the following, unless otherwise specified, we will adopt choice (B).

3.2 Computational uncertainties

First we examine the computational uncertainties. We expect these to be small because we compute only the linear periods of the fundamental and first overtone. In contrast to the linear growth-rates, the periods are very insensitive to the convective parameters ($\alpha$'s in Yecko et al. 1998). The comparison of purely radiative models with our turbulent convective ones gives an idea of the uncertainty. We find that the period shifts are systematic but small, of the order of the size of the dots in Figs. 1 and 2. The fact that they are systematic indicates that they cannot contribute to the scatter of Figs. 1 and 2. The models have been computed with a mesh of 200 points. Models run with a cruder mesh distribution give essentially the same M-L picture. We can safely use linear periods, because nonlinear hydrodynamic modeling shows that the differences are systematic and at most of the order of 0.1% which has no appreciable effect on the M-L picture.

We have not been able to find a computational uncertainty that can account for the scatter in the M-L relation, and we have to look in the observational data.

3.3 Scatter in the $\mathsfsl{M}$- $\mathsfsl{L}$ relations

In the upper panel of Fig. 3 we plot the residuals of the period-luminosity (P-L) relation in V and I for both the LMC and the SMC fundamental Cepheids OGLE data. These diagrams illustrate the structure of the Cepheid P-L relation (see Figs. 5 and 6 from Sasselov et al. 1997). The dispersion is mainly along the reddening vector in the LMC, whereas in the SMC it is not because depth effects are another source of scatter. We recall (e.g., Sasselov et al. 1997) that there is an unfortunate near degeneracy between lines of constant period and reddening. Therefore one cannot just minimize the dispersion along the reddening vector in this plane to correct for the reddening. It would lead to an overcorrection. When we use the reddening derived by U99 we note a very marginal improvement of the residuals as shown in the lower panel of Fig. 3 and as noted by U99.

We conclude that the differential reddening within the clouds on a star by star basis persists as a major source of dispersion that is not compensated for by the reddening maps from red clump stars.

In order to see whether the size of error that is inherent in the observations is responsible for the scatter in our M-L relation we have made the following test. First we construct a sequence of fundamental Cepheid models with a specific M-L relation, ${{\rm Log}}~L = 0.79 + 3.56~{{\rm Log}}~M$, and with a range of $T_{\rm {ef f}}$ that spans the corresponding instability strip. With Eqs. (1-6) we transform L and $T_{\rm {ef f}}$ to I and V magnitudes, and maculate these data with Gaussian noise of intensity 0.02 in the I and V magnitudes, and with a Gaussian noise in the reddening with $\sigma_{E(B-V)}$ = 0.06. Using these surrogate stars as input we then proceed to compute the surrogate stellar masses the same way as we handled the OGLE data. Figure 4 shows the resulting M-Lrelation. It is seen to exhibit the same type of scatter as the OGLE derived M-L relations.

This sensitivity can also be seen analytically. For that purpose we have made a rough fit with the help of our models

$${{\rm Log}}~ P_0 = 11.80 - 0.595 {{\rm Log}}~ M + 0.82 {{\rm Log}}~L - 3.55 {{\rm Log}}~{T_{\rm {ef f}} }.$$ (7)

Together with Eq. (6) one then derives

$$\delta {{\rm Log}}~M 0.51 \delta \mu - 0.51 \delta {V} +1.37 \delta ({V-I})$$ = (8)

$$\delta {{\rm Log}}~L 0.4\delta \mu -0.4\delta{V} +1.32 \delta {E(B-V)}.$$ = (9)

As a check, with the noise level 0.02 in the I and V magnitudes and 0.06 in the reddening for the preceding test one obtains $\delta{{\rm Log}}~ M = 0.05$ and $\delta{{\rm Log}}~ L = 0.09$, in agreement with the numerical results.

The reason for the scatter in the derived M-L relation is thus seen to originate in the extreme sensitivity of the mass-luminosity relation to small photometric errors in the V and I magnitudes.

It is very tempting to use the observational deviations in Fig. 3 to tighten the derived M-L data. The question is whether we can use the deviations parallel to the reddening line to estimate (and correct for) the reddening and observational noise. We find that, because of the finite width of the instability strip, the spread in $T_{\rm {ef f}}$ has an effect parallel to the reddening, so we cannot decouple the reddening error from it. The spread in mass (for a given L) has an effect not parallel (at an angle close to 45 degrees) to the reddening, thus with a component perpendicular to the reddening line. Because of this projection angle the perpendicular direction alone cannot be used to estimate the observational errors in I or V. In summary, it is unfortunately not possible to use the residuals of Fig. 3 to correct for the observational reddening errors on a star by star basis.

Figure 4: Mass-luminosity relations for surrogate stars.

Figure 5: Theoretical HR diagram and mass-luminosity relations for SMC (the two left panels) and for LMC (the two right panels). As in Fig. 1, fundamental Cepheids are shown as solid and overtones as open circles calculations: Theoretical HR diagram with superposed evolutionary tracks from Girardi et al. (2000). M-L relations from evolutionary calculations; solid lines: Girardi et al. (2000), dotted lines: Alibert et al. (1999), dashed lines: Bono et al. (2000).