4 Discussion

The progression in $\phi _{21}$ outlined in Fig. 2 by the outer disc Cepheids is very sharply defined for 3<P<8 days, showing both that the Fourier parameter determination is very accurate, and that the phase shift depends essentially only on the pulsation period. The sequence defined by our metal-deficient targets is indistinguishable from the solar-neighbourhood sequence for low periods, indicating a negligible metal dependence in this regime. This tends to confirm the indication from photometric sequences, which do not show a metallicity dependence either (Udalski et al. 1999a), and is in marginal conflict with the models of Buchler (1997) which predict a dependence of about 0.2 rad in this period range.

The fact that the period-$\phi _{21}$ relation is so tight also indicates that the instability strip is very narrow in this period range (Buchler et al. 1990), possibly narrower than for solar metallicity.

The semi-amplitude A₁ and amplitude ratio R₂₁ are significantly higher for metal-deficient Cepheids at the shortest periods, P<5 days, than for solar-metallicity Cepheids. The excess of R₂₁ is in qualitative agreement with the predictions from the Buchler (1997) models. Our Cepheids are also seen to reach a lower minimum period than in the solar neighbourhood, a decrease related to metallicity effects on stellar evolution, abundantly documented in the LMC and the SMC microlensing data (Baraffe et al. 1998; Bauer et al. 1999).

The higher scatter in the A₁ and R₂₁ amplitudes - compared to the $\phi _{21}$ sequence - is real given the very small uncertainties on these numbers for our programme stars. This scatter is also observed in the solar neighbourhood sample, and is thought to reflect a dependence of pulsation amplitude at a given period on the exact location of a Cepheid in the instability strip (Bono et al. 2000).

Our sample unfortunately contains no object in the period range 8-12 days, near the P₂/P₀ resonance, where metallicity effects would be easiest to detect. The reason for this absence is simply that all objects in this period range in Pont et al. (1997) are situated in the northern hemisphere, out of reach of the Euler telescope at La Silla. These objects - AD Cam, AA Gem, HZ Per, SV Per - should be considered for future radial velocity observations from the north.

The only long-period object in our sample, HW Pup at P=13.45 days, is situated on the other side of the resonance. Its $\phi _{21}$ parameter is markedly higher than the solar-neighbourhood value, which could indicate a metallicity effect.

However, the possibility of HW Pup being a type II Cepheid should first be considered. Type II Cepheids are stars both older and less massive than classical Cepheids, that are about 1.5 mag fainter and have slightly different pulsation curve shapes. Their kinematical properties are typical of the thick disc, and they are usually detected by their high galactic latitude. Fernie & Ehlers (1999) have shown that, for periods between 10 and 20 days, type II Cepheids have a larger $\phi _{21}$ phase shifts, at a given period, than classical Cepheids (see Fig. 1 of Fernie & Ehlers 1999). Although their study was based solely on light curves, a similar effect can be expected on the velocity curve, and HW Pup possesses a $\phi _{21}$ parameter compatible to the type II locus.

In order to check whether the parameters of HW Pup are typical or standing out among other outer disc Cepheids, we examined the photometric data. The V-band light curve of 8 outer disc Cepheids from Pont et al. (1997) with periods between 12 and 26 days were decomposed in Fourier coefficients (the eight objects are CY Aur, YZ Aur, BM Per, HZ Per, OT Per, HW Pup, VZ Pup and AD Cam). The results, displayed in Fig. 3,

Figure 3: Photometric phase shift $\phi _{21}^{\rm phot}$ for: filled circles - Outer disc Cepheids. Open circles - Outer disc Cepheids with mean Vr more than 25 kms-1 from a flat Galactic rotation curve at $v_{\rm circ.}=220$ kms-1, namely BM Per (P=22.96 day) and VZ Pup (P=23.17 day). The photometric data were taken from Moffett & Barnes (1984) and Berdnikov (1986, 1992). Asterisks - Solar neighbourhood Cepheids according to Antonello & Morelli (1996) and Moskalik (priv. comm.). Diamonds - LMC Cepheids from the OGLE survey (Udalski et al. 1999a).

show that the photometric $\phi _{21}$ of HW Pup is perfectly normal for an outer disc Cepheid, and that indeed outer disc Cepheids as a whole have systematically larger $\phi _{21}$ than solar-neighbourhood Cepheids in the period interval between 12 and 20 days, higher by about half a radian. For these 8 objects, the residuals of the center-of-mass radial velocities around a constant rotation curve at $v_{\rm circ}=200$ kms^-1 have a dispersion of 12.1 km$\,$s^-1. Such a small velocity dispersion, characteristic of the young thin disk, excludes an important contamination by type II Cepheids. It is therefore concluded that HW Pup is very probably a genuine classical Cepheid, not a type II Cepheid, and that its higher $\phi _{21}$ in radial velocity is typical of outer disc Cepheids in that range of period, indicating a metallicity effect.

A posteriori, a hint of the same effect can be seen in the LMC photometry from the OGLE survey (Udalski et al. 1999a). We have Fourier decomposed the V lightcurves of OGLE LMC Cepheids and their $\phi _{21}$ parameters are shown in Fig. 3. The $\phi _{21}$ excess between 12 and 20 days visible in LMC data is compatible with the position of our outer disc objects, $\sim$0.5 radian above the solar metallicity sequence. SMC Cepheids (Udalski et al. 1999b) and IC 1613 objects (Antonello et al. 2000) do not show a further increase of $\phi _{21}$. If due to metallicity, the increase appears to occur mostly in the metallicity range between solar and around half-solar with little or no further change below half-solar.

Interestingly, such a metallicity effect is reminiscent both in direction and amplitude of the model predictions of Buchler (1997). The models predict a rapid increase of $\phi _{21}$ for half-solar metallicity and periods above 10 days.

To confirm this conjecture, velocity curves for outer disc and LMC Cepheids should be gathered in the 8-20 days period range. Our measurements provide a precise definition of the Fourier parameter progression in the 3-8 day period range, and a solid suggestion of a $\phi _{21}$ excess around P=13 days. The next step is to extend this work near and above the resonance period. If our tentative scenario is correct, we would expect a significantly smaller resonance peak in the $\phi _{21}$ - period relation than for solar-metallicity Cepheids. Apart from the four northern outer disc Cepheids mentioned above, the natural candidates are LMC and SMC Cepheids. Up to now, there is little velocity data for Magellanic Clouds Cepheids, but several studies are in progress (Gieren et al. 2000; Storm et al. 2000) that may soon yield well-covered velocity curves in this period range.

Finally, we point out the interesting possibility that the common $\phi _{21}$excess of local type II Cepheids (Fernie & Ehlers 1999) and outer disc Cepheids for P=10-20 days may not be a coincidence: the increased phase shift for type II Cepheids could also be caused by metal deficiency, type II Cepheids belonging to a thick disc population markedly more metal-poor than local classical Cepheids. The metallicity effect on pulsation curve morphology may thus be similar regardless of Cepheid type.