3 Fourier parameters

The velocity curves were decomposed in Fourier sine series of 3rd to 6th order by unweighted least-square fitting, as described in Kienzle et al. (1999). The amplitudes A_k and the phases $\phi_{k}$ of the harmonics were combined according to the usual definition of the Fourier parameters to form the amplitude ratios R_k1=A_k/A₁ and the phase shifts $\phi_{k1}=\phi_{k}-k\phi_{1}$. In all cases, the number of points and careful phase coverage allowed a very accurate determination of the first Fourier parameters, $\phi _{21}$, A₁ and R₂₁.

The points with residuals higher than 2.5 $\sigma$ from the fitted curve were removed (crosses in Fig. 1). All these points are Coravel measurements, with higher observational uncertainties. As Fourier decomposition may perform poorly in pulsation curves with a linear ascent followed by a steep descent (periods near 4.5 days), we added interpolated points in the ascending part of the velocity curve in two cases - FG Mon and WW Mon - in order to avoid unrealistic excursions of the fitted curve. The period was either fixed to the value quoted in the General Catalogue of Variable Stars (Kholopov et al. 1998) or left as a free parameter, depending on the quality of the phasing of the radial velocity data. The period of WW Mon was fixed at 4.66221 days (while $P_{{\rm GCVS}}=4.66231$ days) in order to phase Elodie and Coralie data. HW Pup data were supplemented with the points of Metzger et al. (1992) to fit the period. Only our measurements were then used to derive the Fourier parameters. Two stars (namely FG Mon and FI Mon) show a significant velocity shift between Coravel and Coralie data sets, probably due to binarity. This is not unexpected in a sample of eleven Cepheids, given the frequency of Cepheids in binary systems (e.g. Gieren 1982; Szabados 1996). In both cases Coravel data have been shifted, by 3.15 km$\,$s^-1 (FG Mon) and 1.3 km$\,$s^-1 (FI Mon), with respect to Coralie velocities in order to match the latter points.

The resulting Fourier parameters are given in Table 1,

 

 

Table 1: Fourier parameters for outer disc Cepheids, with associated uncertainties. M, N and $\sigma$ are, respectively, the number of datapoints, the order of the fit and the standard deviation of residuals. The uncertainty on the period is indicated only when it was a free parameter. A₀ corresponds to the center-of-mass radial velocity, A₁ is the semi-amplitude of the first order, R₂₁ the amplitude ratio of the two first orders, and $\phi _{21}$ their phase shift. $R_{\rm GC}$ is the galactocentric distance, and [Fe/H] the metallicity calculated from $R_{\rm GC}$ using a radial metallicity gradient of -0.07 dex/kpc.

Star	Period [d]	M	N	$\sigma$ [kms-1]	A0 [kms-1]	A1 [kms-1]	R21	$\phi _{21}$	$R_{\rm GC}$ [kpc]	[Fe/H]
HW Pup	13.457423	30	5	0.728	117.456	19.877	0.124	4.246	13.8	-0.40
	-				0.152	0.232	0.012	0.081
V510 Mon	7.457428	20	3	0.481	63.424	9.879	0.421	3.594	12.5	-0.30
	0.000111				0.112	0.151	0.018	0.053
TZ Mon	7.428134	37	5	0.554	53.933	14.921	0.552	3.619	11.4	-0.25
	0.000019				0.112	0.163	0.014	0.033
TW Mon	7.097064	24	5	0.388	82.938	12.887	0.508	3.510	13.6	-0.40
	0.000089				0.087	0.129	0.014	0.027
XX Mon	5.456543	22	5	0.899	66.383	15.555	0.432	3.253	12.1	-0.30
	0.000016				0.207	0.310	0.023	0.060
CU Mon	4.707547	20	5	0.962	61.542	17.007	0.442	3.078	13.5	-0.40
	0.000083				0.256	0.335	0.025	0.064
WW Mon	4.662210	24	6	1.790	52.566	18.327	0.492	3.046	12.2	-0.30
	-				0.370	0.543	0.034	0.084
FG Mon	4.496590	24	6	1.178	91.501	17.214	0.460	3.019	13.6	-0.40
	-				0.245	0.340	0.023	0.066
BC Pup	3.544217	26	6	1.109	90.787	18.725	0.441	2.916	12.9	-0.35
	0.000051				0.277	0.356	0.022	0.062
FT Mon	3.421740	28	6	1.421	55.147	19.351	0.403	2.946	13.0	-0.35
	-				0.302	0.424	0.026	0.072
FI Mon	3.287822	25	5	0.580	86.105	18.656	0.370	2.950	12.2	-0.30
	-				0.136	0.183	0.011	0.041

and the progression with period of the $\phi _{21}$, A₁ and R₂₁ parameters is illustrated in Fig. 2.

Figure 2: Radial velocity semi-amplitude A1 (km$\;$s-1), phase shift $\phi _{21}$ (rad) and amplitude ratio R21 ( top, middle and bottom plot respectively) versus pulsation period. Asterisks - Solar neighbourhood sample from Moskalik et al. (1999). Dots - Our outer disc sample. Error bars are plotted only when larger than symbol size. The theoretical progression according to the 2:1 resonance scenario is shown as a solid line. The resonance period ($P_{\rm r}$) has been fixed to 10.003 days (P. Moskalik, priv. comm.). As the theoretical amplitude ratio depends on the artificial viscosity parameter, it can be scaled by an arbitrary factor. In the R21 plot a scaling factor of 0.77 has been applied (dotted line) to the original curve (solid line). The isolated point at P=13.46 days is HW Pup.

The outer disc Cepheid data are plotted as dots along with solar-neighbourhood data (asterisks) from Moskalik et al. (1999).