1 Introduction

Already in mid-century, the technique of cross-correlation was known in the field of signal-processing as it was implemented in radar-receivers to measure time-delay (distance). A rigorous discussion of the precision of such measurements was given by Woodward & Davies (1950) (see also Woodward 1953). Based on these results a similar technique for the measurement of Doppler shifts in astronomy was proposed by Fellgett (1953) and developed in practice by Griffin (1967). Since then it has been widely used and discussed in the astronomical literature (see e.g. Tonry & Davis 1979; Connes 1985 and references therein). A major turning point was the advent of digitized spectra (and computers to handle them) which made it possible to represent a spectrum as a function of log(wavelength) so that a stellar Doppler-shift measurement became the perfect analogue of the measurement of time-delay in radar. Significant differences with the case of signal-processing in radar are that the "signal" in our case is not an isolated pulse, and that in general a variation of the noise level with wavelength has to be taken into account (see also Fellgett 1953).

Cross-correlation techniques for the derivation of stellar radial velocities have so far been applied most successfully to late-type spectra. These contain numerous, sharp, and useful lines which guarantee a narrow well-defined cross-correlation peak whose centering is straightforward (e.g. Scarfe et al. 1990; Latham 1992; Baranne et al. 1996). Precisions (random errors) and accuracies (systematic errors) in the range 0.1-1 km s^-1 are routinely obtained from these spectra. The situation for early-type spectra, however, is different at three levels. First, owing to the low line density and typically high rotational velocity, the random error on a radial velocity shift is much higher for a given signal-to-noise and a given wavelength span of the spectra. Secondly, the cross-correlation peak in general is much broader due to the occurrence of intrinsically broad lines (H and He) and due to rotation; in addition, it contains important sub-structure caused by the mixing of spectral lines of different width; both complicate its accurate centering. Figure 1 shows an example of the differences occurring between early and late-type spectra. Thirdly, also the accuracy of the measured radial velocity shift is lower due to, on the one hand, a much more likely (and more harmful) spectral mismatch between object and template, and on the other hand, the occurrence of spectral lines influenced by atmospheric velocity fields. A more detailed discussion of the problems related to radial velocity work on early-type stars can be found in Verschueren (1995). It is important to realise, however, that some of the problems just mentioned are also likely to occur for late-type spectra if a precision and accuracy one or two orders of magnitude higher would be aimed at (e.g. due to convective shifts).

  

Figure 1: a,b) Synthetic early-type spectrum with a continuum S/N = 50, and the central part of its cross-correlation function with an intrinsically identical template of S/N = 200. c,d) Idem for a solar spectrum

Improving the precision and accuracy of radial velocities of early-type stars as much as possible (in particular getting below the 1 kms^-1 level) is important for a variety of astrophysical applications, such as kinematical and dynamical studies of young stellar groups, detection of early-type binaries, and correction of high-precision proper motions for perspective acceleration. The first results of a study to improve the accuracy for early-type stars by minimizing spectral-type mismatch is presented elsewhere (Verschueren et al. 1999a,b). Here, we address the minimization of random errors (due to photon and read-out noise) in the absence of any intrinsic object-template spectral mismatch; this is related to the problems of the first and second level mentioned above.

In Sect. 2, we derive a theoretical lower bound to the random error on a radial velocity shift, taking into account random noise on both object and template spectrum. Section 3 shows which specific cross-correlation function fitting techniques are required to minimize the random error in practice, while Sect. 4 discusses the influence of rotational mismatch on the random error attained. Section 5 deals with the estimation of the actual random error on a measured Doppler shift, i.e. taking account of the actual way of extracting the information. Section 6 lists the conclusions.