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Astron. Astrophys. 318, 416-428 (1997)

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3. Distance determination

The 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 [FORMULA] (E(V-I)/E(B-V)), as the angle between the reddening vector and the intrinsic locus is small (less than [FORMULA] for short periods, up to about [FORMULA] for longer periods, Fig. 4). In fact the combined effects of high reddening and low metallicity make this method very fragile for short-period cepheids. For the PL relations with PC reddenings on the contrary, the metal dependence can be checked with Magellanic Clouds data, and the reddening determination is not affected by the value of [FORMULA] or by the amount of reddening (see Fig. 5 and Appendix). Moreover, using a PL relation with PC reddenings takes advantage of the fact that the colour term in the PLC relation is numerically similar to R (reddening to absorption ratio) and the two deviations from the ridge line tend to cancel each-other in their effect on the inferred distance (Appendix, Eq. 3).

[FIGURE] Fig. 4. Colour-colour diagram for the sample: (B-V) vs. (V-I). The points are the measured means, the line the intrinsic locus (Dean et al. 1978). The dotted line is the locus corrected for metallicity effects, using [Fe/H]=-0.5. The arrow is the reddening vector. Bringing the measured colours back on the intrinsic locus using the reddening vector is the "colour-colour" (CC) method of reddening determination. Note that for small periods (small [FORMULA]), the angle between the reddening vector and the locus becomes small, and metallicity effects are amplified.

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.

[FIGURE] Fig. 5a and b. Period-colour (PC) diagrams for the sample. Left: B-V, right : V-I. The line is the ridge line of the cepheid locus. The dotted line is the relation for [Fe/H]=-0.5. The reddening vector is vertical by definition. Note that the V-I relation is less sensitive to metallicity, and that the uncertainty is not affected by the amount of reddening (see text).

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): [FORMULA] and -0.185 (SMC) (45 and 47 objects resp., same source). Assuming [FORMULA] and [FORMULA] for the clouds (Feast 1986 and Caldwell & Coulson 1986), we fit the following metallicity corrections on Clouds data:

[EQUATION]

Then

[EQUATION]

We apply the PL relation from the review by Feast & Walker (1987):

[EQUATION]

The bolometric relation is assumed to be independent of metallicity, but the bolometric correction applied to get the PL relation in [FORMULA] is metallicity dependent. The correction in Caldwell & Coulson (1987) is used:

[EQUATION]

With (Feast & Walker 1987)

[EQUATION]

we get the distance modulus:

[EQUATION]

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):

[EQUATION]

3.2. Distances using (V-I) and I

Similarly, we use the relations:

[EQUATION]

fitted on Magellanic Clouds data from Caldwell & Coulson (1985a).
With the PL relation of Caldwell & Coulson 1987:

[EQUATION]

and (Dean et al. 1978):

[EQUATION]

one gets

[EQUATION]

3.3. Choice of distance scale

The 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 [FORMULA]) 1.

3.4. Metallicity gradient

Throughout 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 pulsators

The 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.
In our sample, only one star, V510 Mon, was seen to have an abnormally low pulsation amplitude or sinusoidal light curve, though its period of 7 days makes it an unlikely candidate.
The rarity of overtones in the sample could be explained by the decrease in overtone frequency with decreasing metallicity (indicated by SMC data e.g. Buchler & Moskalik, 1994) and a detection bias favoring large amplitudes.

3.6. Type II cepheids

It 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:
[FORMULA] high z coordinate. The z coordinates were corrected for the fact that the cepheid plane is lower and slightly tilted relative to the galactic plane (see for instance Fernie 1995). Objects were tagged if farther than 300 pc from the plane [FORMULA] pc (d is the distance from the sun in kpc).
[FORMULA] low reddening compared to their distance.
[FORMULA] low (P [FORMULA] 3 d) or high (P [FORMULA] 10 d) period. The distribution of type II cepheids peaks at low ("BL Her" objects) and high ("W Vir" objects) periods compared to classical cepheids.
[FORMULA] position. We tagged stars isolated from any other classical cepheid.
[FORMULA] objects labeled as "CEP" (cepheid of undetermined type) in the GCVS.

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 Galaxy

Fig. 6 and 7 display the position of the sample cepheids in the Galactic plane and in z, using distances determined here.

[FIGURE] Fig. 6. Position of cepheids in the galactic plane. X and Y are coordinates in kpc. The Sun is at the origin, the Galactic Centre at the bottom. Empty symbols indicate cepheids previously studied for galactic kinematics, solid symbols the cepheids of this sample. Circles are indicated at R= [FORMULA], R=11.5 kpc and R=15 kpc.

[FIGURE] Fig. 7. Position of the cepheids in z relative to the "mean cepheid plane" defined as [FORMULA] pc, as a function of l. The lack of objects near z=0 is due to the high absorption in the plane.
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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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