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Astron. Astrophys. 318, 416-428 (1997) 3. Distance determinationThe distance of classical cepheids can be known with a high accuracy using period-luminosity (PL) and period-luminosity-colour (PLC) relations (see for instance the review by Feast & Walker 1987), making them a privileged standard candle. Some care must be taken in applying these relations to outer disc cepheids for two reasons: high reddening and metal deficiency. The objects in our sample have typical reddenings between 0.5 and 1 mag, and metallicities probably ranging between the LMC and SMC values (assuming a metallicity gradient in the disc of -0.05 to -0.10 dex/kpc). In order to obtain reliable distances, it is thus necessary to get precise reddenings, and to allow for metallicity effects on the PL/PLC relations. These two points are related, since if the reddening is determined from the colours of the cepheid itself -as is the case in this study- it is important to take into account the effect of metallicity on the intrinsic colours of the cepheid. The effect of metallicity on cepheid colours and magnitudes is a complex question, still far from being satisfactorily answered. Several semi-empirical corrections have been proposed (Stothers 1988, Caldwell & Coulson 1985a, Freedman & Madore 1990, Stift 1995), but a clear consensus is yet to emerge (see remarks in Feast 1991). Purely theoretical predictions (Chiosi et al. 1993) do not reproduce quantitatively the cepheid colours observed in the Magellanic Clouds. The only point on which an agreement has been reached is the insensibility of the bolometric PL relation to metallicity. We have decided to adopt metallicity corrections derived as much as possible from observations, taking advantage of the fact that the metallicities of the stars in our sample are bracketed between LMC and SMC metallicities, for any reasonable assumption on the metal gradient in the disc. Since many cepheids have been observed in both Clouds -which are only slightly reddened- they provide an important check on colour and magnitude variations with metal deficiency. Two sets of relations can be used to determine the reddening and
distance of a cepheid. The first ones are almost exact, being
consequences of physical constraints on the pulsation: the
colour-colour (CC) relation for reddenings and PLC relation for
distances. The second ones are only average, as a consequence of the
finite width of the instability strip: the period-colour (PC) relation
for reddening and PL relation for distance. It is usual to determine
reddenings with a CC relation, generally B-V vs. V-I, by fitting the
observed colours on an intrinsic cepheid locus (Dean et al. 1978,
Fig. 4). Now, in the case of deficient, reddened objects, the PLC
and CC relations are very tricky to correct for metallicity, and the
reddening determination vulnerable to uncertainties in the correction
and to any change, even slight, in the assumed
In choosing which relation to use, one has to keep in mind that for our purpose any systematic bias is going to affect the resulting rotation curve -especially a bias increasing with distance. The main concern must be to avoid such a bias, even if it means accepting a slightly higher dispersion.
This has led us to use a PC relation for reddening and a PL relation for distance, rather than the usual PLC/CC pair. The Appendix shows, using order-of-magnitude calculations, why for highly reddened, metal-deficient cepheids, PL distances with PC reddenings are much more robust against systematic biases, and not more scattered, than PLC distances with CC reddenings. We give hereafter two distance determinations with the PC/PL combination, the first with B-V as a colour and V as a magnitude, the second with V-I as a colour and I as a magnitude. We assume throughout that the slopes of the PL and PC relations are unaffected by metallicity (Stothers 1988, Chiosi et al. 1993). 3.1. Distances using (B-V) et V
We used the PC relation from Laney&Stobie (1994), based on 46
galactic cepheids, who find the following corrections for the
Magellanic clouds (temperature change + blanketing):
Then We apply the PL relation from the review by Feast & Walker (1987): The bolometric relation is assumed to be independent of
metallicity, but the bolometric correction applied to get the PL
relation in With (Feast & Walker 1987) we get the distance modulus: The above relations were calibrated using intensity averages for (B-V), whereas we computed magnitude averages. The first are transformed into the second using Fernie (1990): 3.2. Distances using (V-I) and ISimilarly, we use the relations: fitted on Magellanic Clouds data from Caldwell & Coulson
(1985a). and (Dean et al. 1978): one gets 3.3. Choice of distance scaleThe second distance determination was preferred when available in the following analysis for the reasons explained in the Appendix, and also because it is less affected by undetected companious (see Sect. 5.3). The Appendix is available electronically at http:://science.springer. Among the 36 stars with both (V, B-V) and (I, V-I) distances, the
dispersion on µ is 0.21 mag, an acceptable value given
the intrinsic width of the instability strip, and the mean shift is
+0.04 mag (in the sense 3.4. Metallicity gradientThroughout the distance determination, a metallicity gradient of the outer disc of -0.07 dex/kpc is adopted (Harris 1981). The effect of changing this assumption is examined in Sect. 5.3 2. 3.5. Overtone pulsatorsThe identification of "s-cepheids", characterized by a sinusoidal
low-amplitude light-curve, with objects pulsating in the first
harmonic mode, has been proposed for some years now (Antonello et al.
1990, Mantegazza & Poretti 1992), and unambiguously confirmed by
results from the EROS (Beaulieu et al. 1995) and MACHO (Welch et al.
1995) projects in the LMC. 3.6. Type II cepheidsIt was realized in 1952 by Baade and others that type II cepheids, though located in the cepheid instability strip, are much smaller, older objects than classical cepheids. It now seems that they do not form a homogeneous population (Harris & Wallerstein 1984). Distinguishing type II cepheids from classical cepheids is particularly tricky in the outer disc. Kinematically, type II cepheids, nearer but lagging behind young disc rotation, may have the same radial velocity as remoter classical cepheids following young disc rotation. The high z coordinate criterion is not a foolproof discriminant either, since in the outer disc, disc thickening and warping may bring classical cepheids farther from z=0. Finally, a relative number of type II cepheids higher than in the solar neighbourhood may be expected for the outer disc sample since, for a given magnitude, type I cepheids are seen at a larger distance, in remoter places of the outer disc where the stellar density is much lower. We have applied five criteria to try to detect possible suspects,
none of them decisive in isolation: Stars with 3 or more of the above criteria were labeled as suspect. The low velocity dispersion of our sample after the subtraction of differential rotation, about 10 km s-1 (see Fig. 10), indicates however that the number of type II cepheids present in the sample is very low (much higher dispersions are expected for type II cepheids, of the order of 30-50 km s-1). The distribution of periods in the sample (Fig. 1) is also typical of classical cepheids and not of type II. 3.7. Position in the GalaxyFig. 6 and 7 display the position of the sample cepheids in the Galactic plane and in z, using distances determined here.
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