6 Conclusions

In order to approach optimal radial velocity precision for early-type stars, a number of precautions have to be taken into account to reduce random errors to levels which are attained effortlessly for late-type spectra.

We generalised the theoretical lower bound for the random error attainable as a function of the intrinsic spectrum morphology and of the wavelength-dependent noise on object and template spectrum (Eq. (4)). It is valid for all (late and early-type) well sampled spectra. It shows that, for other than small rotational velocities, a high S/N and a large wavelength region are required to beat the random errors sufficiently in early-type spectra (Eqs. (7, 8)).

Owing to the complex morphology of a cross-correlation peak derived from blended and broadened early-type spectra, containing mixtures of metal, He and H lines, specific centering techniques are required to approach this lower bound for the random error in practice (even in the absence of systematic mismatch). Based on a wide range of Monte-Carlo experiments with synthetic early-type spectra, an optimal cross-correlation peak centering method is proposed: merely the top of the peak (5 pixels) is fitted with a low-order polynomial (2$^{\rm nd}$ degree), after a well chosen high-frequency Fourier filtering has been applied. The filtering allows to approach very closely the lower bound for the random error using a very small amount of fit-points around the maximum position of the cross-correlation peak; using merely the centre of the cross-correlation peak is desirable because of the fact that there the radial velocity information is least perturbed, and for practical reasons. The optimal filter is computed as a function of the rotational velocity of the spectrum, while it also varies, to a lesser degree, with S/N (Fig. 6).

Monte-Carlo experiments with synthetic spectra show that rotational mismatch, i.e. cross-correlating a rotationally broadened object with a narrow-lined template, does not increase (nor decrease) the minimum random error attainable with a broad-lined template. It is, however, virtually impossible to attain this minimum error in practice since it is connected to a very small range of fit-intervals of an a priori unknown length (Fig. 7). Using a template of roughly equal rotational velocity (and preferably equal inclination angle) as the object spectrum, in combination with high-frequency filtering, is therefore mandatory. Note that vsini matching was also recommended by Peterson et al. (1984) in order to avoid multiple components of (nearly) equal strength on the cross-correlation peak in spectral regions with blended lines.

Finally, we investigated whether the error estimate based on the r-statistic (Tonry & Davis 1979; Kurtz & Mink 1998) can be used to predict the actual random error on a cross-correlation derived radial velocity shift between early-type stars. We showed that the actual relation between the true error $\epsilon$ and the antisymmetric part of the cross-correlation function $\sigma_{\rm a}$, theoretically depends on the structure of the spectrum (Appendix). Owing to their strongly fluctuating nature and their similar appearance, one could nevertheless find a "generally valid" relation between $\epsilon$ and $\sigma_{\rm a}$for late-type spectra. Our Monte-Carlo experiments with different types of early-type spectra, however, show that this is impossible for the latter (Fig. 8). We conclude that its use is inadequate for early-type spectra for both theoretical and practical reasons.

Acknowledgements

We thank Herman Hensberge for many stimulating discussions, Myriam Vrancken for kindly providing the synthetic spectra, and the referee for valuable comments. WV acknowledges substantial financial support from the Fund for Scientific Research - Flanders (Belgium) (F.W.O.) through Research Grant No. 1.5.549.98.