1 Introduction

In the last few years, high quality data on large numbers of Cepheid variables in the Small and Large Magellanic Clouds have been made available by the EROS and OGLE microlensing projects (Beaulieu et al. 1995; Afonso et al. 1999; Udalksi et al. 1999a-c). In particular the OGLE Project has provided standard colors in addition to periods and magnitudes for the largest samples published to date. In this paper we examine some of the constraints that the MC Cepheids impose on stellar evolution and stellar pulsation theories.

Figure 1: Mass-luminosity relations for SMC and LMC as derived from the OGLE data adopting choice a) for distance modulus and reddenings in the left panel and our preferred choice b) in the right panel; fundamental Cepheids are shown as dots and overtones as open circles. As a guide to the eye we have superposed the evolutionary M-L relations of Girardi et al.

We use the full catalogue of publicly available of LMC and SMC single mode Cepheids and SMC double mode Cepheids produced by OGLE in BVI (Udalski et al. 1999a-c, with the zero point corrections as suggested in the April 2000 OGLE web site, U99 hereafter). The single mode Cepheid catalogues contain 1435 LMC and 2167 SMC stars. We keep the objects classified as fundamental mode pulsators or first overtone pulsators, with reliable photometry in both V and I. We exclude stars whose magnitudes are most likely to be strongly contaminated by companions or blending in V or I. The remaining stars form our working sample of OGLE Cepheids. It consists of 670 LMC fundamentals, 426 LMC overtones, 1197 SMC fundamentals and 677 SMC overtones, as well as 24 F/O1 SMC double-modes and 71 O1/O2 SMC double-modes.

The OGLE data base provides intensity averaged magnitudes and colors. With the help of distance moduli, these can then be transformed to luminosities and effective temperatures. The Magellanic Clouds (MC) are thought to be relatively uniform in composition, and with the observed average compositional information theoretical modeling can then provide the mass of each star.

We have to adopt distances and reddenings for the Magellanic Cepheids. The distance of the Large Magellanic Cloud remains at the center of the current debate about the distance scale ladder. Extreme values of the distance modulus range from 18.08 to 18.70, but the distance estimates tend to cluster around 18.3 for the "short'' distance scale, and around 18.5 for the "long'' distance scale (see Walker 1999; Udalski 2000 and references therein; Cole 1998; Girardi 1998; Stanek et al. 2000; Romaniello et al. 2000; Feast & Catchpole 1997; Luri et al. 1998; Groenewegen & Oudmaijer 2000 and references therein; Groenewegen 2000; Sakai et al. 2000; Cioni et al. 2000). It is beyond the scope of the paper to solve this distance scale problem, but we will justify the choices we adopt for distances and reddenings.

U99 derive reddening by making the assumption that the red clump Iluminosity has a weak metallicity dependence, and they use the Iluminosity to map relative variations of E(B-V). Finally they fix the zero point of the relative scales on the basis of observations of NGC 1850, NGC 1835, HV 2274 in the LMC, and NGC 416 and NGC 330 in the SMC. These assumptions lead to the determination of a "short'' distance to the LMC of $18.24 \pm 0.05$ and $18.75 \pm 0.05$ to the SMC. The relative distance between LMC and SMC obtained with four different distance indicators (Cepheids, RR Lyrae, red clump and tip of the red giant branch) but with the same reddening maps, gives a relative distance modulus of $0.50 \pm 0.02$.

The reddening maps of U99 give a mean reddening for the LMC of 0.147, and of 0.092 for the SMC. The consistency of the zero point of the reddening scale between these different observations is at the level of few times 10^-2 (e.g., $E(B-V)=0.15 \pm 0.05$ for NGC 1850 and $E(B-V)=0.13 \pm 0.03$ for NGC 1835). We note that these values are different from what is usually given as mean properties for the clouds (especially for the LMC, see Walker 1999 and references therein). In his recent review, Walker (1999) noted that the median reddenings are $E(B - V) \sim 0.1$ for the LMC and $E(B - V) \sim 0.08$ for the SMC. The galactic foreground reddenings along the line of sight of the clouds are known to be low, viz. 0.06 and 0.04, respectively. The estimation of differential reddening inside the clouds based on earlier studies is quite uncertain. In particular, heavily reddened stars ( E(B- V)=0.30) can be found all over the LMC, but the typical range is -0.15.

In view of these uncertainties and controversies, we consider three alternative choices for the distance moduli and for the reddening corrections for our derivation of stellar parameters from the OGLE data:
Choice (A) adopts both the distance moduli and the reddening as suggested by U99.
Choice (B) adopts the Cepheid distance modulus to the LMC of $18.55\pm0.10$, and a relative distance between LMC and SMC of $0.42\pm 0.05$ (Laney & Stobie 1994), with the mean reddenings of E(B- V)=0.1 and E(B- V)=0.08 for LMC and SMC respectively.
Choice (C) is intermediate in that it adopts U99's relative distance between LMC and SMC of $0.50 \pm 0.02$ and its reddening, but a "long'' LMC distance modulus of 18.55.

To summarize the differences, (B) - (A), in distance moduli and in mean reddenings are $\delta \mu = 0\hbox{$. ^{\rm m}$ } 31$, $\delta \langle E(B - V) \rangle \sim 0.047$ for the LMC, and $\delta \mu = 0\hbox{$. ^{\rm m}$ } 22$, $\delta \langle E(B - V) \rangle \sim 0.012$ for the SMC. The differences, (C) - (A), in distance moduli and in mean reddenings are $\delta \mu = 0\hbox{$. ^{\rm m}$ } 25$, $\delta \langle E(B - V) \rangle = 0$ for the LMC and the SMC.

Figure 2: Mass-luminosity relations for SMC and LMC as derived from the OGLE data with choice c) for distance modulus and reddenings. Superposed are the M-L relations of Girardi et al.

We follow Kovács (2000) in the conversion from magnitudes to bolometric, and from colors to effective temperatures. Because of the wider range of masses that are needed for this study, we have redone the Kovács fits with the Kurucz (1995) stellar atmospheres from the BaSeL database (Lejeune et al. 1997) for the LMC and SMC chemical compositions for the temperature, luminosity and $\log g$ range appropriate for Cepheids ranging from 100 $L_\odot$ to 20000 $L_\odot$, with the results

 

$$2.5\thinspace {{\rm Log}}\thinspace L \mu_{\rm MC}- V +R_V\thinspace E(B\negthinspace -\negthinspace V) + BC +4.75$$ = (1)

$${{\rm Log}} \thinspace g 2.62 - 1.21\thinspace {{\rm Log}} \thinspace P_0$$ = (2)

$${\rm SMC\hfill} :$$

$${{\rm Log}} \thinspace{T_{\rm {ef f}} } 3.91611 + 0.0055\thinspace {{\rm Log}} \thinspace g$$ =

$$\quad - 0.2482 (V\negthinspace -\negthinspace I_{\rm c} \negthinspace -\negthinspace (R_V-R_I) E(B\negthinspace -\negthinspace V))$$ (3)

$$-0.0324 + 2.01 \Delta T - 0.0217\thinspace {{\rm Log}} \thinspace g$$ BC =

$$\quad - 10.31\thinspace \Delta T^2$$ (4)

$${\rm LMC\hfill} :$$

$${{\rm Log}} \thinspace{T_{\rm {ef f}} } 3.91545 + 0.0056\thinspace {{\rm Log}} \thinspace g$$ =

$$\quad - 0.2487 (V\negthinspace -\negthinspace I_{\rm c} \negthinspace -\negthinspace (R_V-R_I) E(B\negthinspace -\negthinspace V))$$ (5)

$$-0.0153 + 2.122 \Delta T - 0.0200\thinspace {{\rm Log}} \thinspace g$$ BC =

$$\quad - 11.65\thinspace \Delta T^2$$ (6)

where $\Delta T = \log {T_{\rm {ef f}} } - 3.772$ and L is in solar units. The transformation to absolute luminosities is then made with the adopted distance moduli $\mu_{\rm MC}$ to the LMC or the SMC for the various choices (A), (B) and (C). We note that compared to Kovács there are systematic shifts of 50 K in $T_{\rm {ef f}}$, and 0.02 in log L.

It may be objected that since $\log g = \log (G M ) /R^2$ in Eq. (2) there is an implicit assumption about a mass-radius -period relation. However, in Eqs. (3-6), $\log g$ appears with a tiny multiplier, and over the period range from 1 to 10 days its contributions to Log L and Log $T_{\rm {ef f}}$ vary by 0.013 and 0.0046, respectively. Ultimately, there is essentially no feedback on our mass determination.

Figure 3: The two upper panels gives the residuals of the V and IP-L relation for uncorrected reddening in LMC and SMC. The two lower are giving the residuals of the V and I P-L relation after reddening corrections based on the red clump stars following the U99 procedure. On each panel are indicated in solid the reddening line, and in dash the depth dispersion (the diagonal) line.