1 Introduction

Techniques to determine radial velocities (RVs) of stars by measuring the Doppler shifts of their spectral lines have continuously improved throughout the last half-century. Cross-correlation techniques that use optical templates (e.g. Baranne et al. [1979]) or numerical ones (e.g. Scarfe et al. [1990]; Latham [1992]; Baranne et al. [1996]) are particulary efficient in that they combine all spectral information and therefore optimize the measuring precision (random errors). There is, however, no guarantee that the accuracy (systematic errors) is optimized as well in this way. Cross-correlations of spectra give rise to systematic errors through "spectrum mismatch'' between object and template spectrum. More specifically, such errors arise from asymmetrical differences, when (for example) the components of a blended line do not have the same relative strength in both spectra. In fact, spectrum mismatch is all but inevitable except in the singular case when both spectra arise from the same non-variable star (and provided that neither the observations nor the data reduction process introduced significant differential error).

Spectrum mismatch is caused by differences in atmospheric parameters ( $T_{\rm eff}$, logg, abundances), rotational velocity, atmospheric velocity fields (e.g. convection, wind, pulsation), and all peculiarities which individual stars or specific groups of stars may exhibit (e.g. magnetic-field structure). If the template is a synthetic spectrum, the amount of spectrum mismatch will obviously depend on the degree of sophistication included in the model, on the accuracy of the line data, and on the amount of information one has about the object; even the best present-day spectrum synthesis does not simulate all the fine details of an observed spectrum. Incidentally, if RVs are determined not by cross-correlating spectra but by measuring the centroids of individual features, spectrum mismatch causes small wavelength offsets in those measured centroids.

For most categories of late-type (F-G-K) stars, radial velocities are routinely obtained nowadays with precisions and accuracies of the order of 0.1 - 1 kms^-1. That success derives from intrinsic characteristics of late-type spectra, namely a high density of lines, small line widths, and lines which generally bear a family resemblance to one another over a wide range of spectral types. Thus, one can achieve small random errors from spectra with relatively low signal-to-noise ratio (S/N) and with a small wavelength coverage. Furthermore, systematic mismatch errors between those spectra are likely to be much smaller than 1 kms^-1 because statistically the effects of the numerous individual blends are approximately evenly distributed in sign and so have a very small nett effect. The same template spectrum can therefore be used for all those late-type stars without the risk of systematic errors, caused by object-template mismatch, larger than quoted above. The latter has facilitated not only the derivation of accurate relative RVs of different late-type stars, but also the definition of a late-type RV standard star system of which the absolute zero-point could be tied to the Sun - the only star whose RV can be measured very accurately independently of its spectrum (Petrie [1962]). Nevertheless, it has recently become clear that, even for late-type stars, RV errors within the aforementioned range are dominated at least in some cases by object-template mismatch; this is instanced by the zero-point offset for the RVs of red stars measured with CORAVEL (Stefanik [1997]). Convective line shifts may also introduce systematic offsets of several tenths of a kms^-1 between different late-type stars (Dravins [1985]). It is also well known that the quoted accuracies cannot be obtained routinely for rotating or very peculiar late-type stars, nor for M-type stars, because of the lack of suitable template spectra.

In terms of absolute (random and systematic) errors, the situation is much worse for early-type (O-B-A) stars. A discussion of the quantification and minimization of random errors that arise when cross-correlating early-type spectra has been given by Verschueren & David ([1999]). The present series of papers undertakes the (harder) task of addressing systematic errors, which are in general much larger than for late-type stars for the following reasons. The total range of temperature spanned by O-B-A stars is much larger than that encompassed by F-G-K stars, and the relative strengths of the components of many blends vary faster from one sub-class to the next. The low number of available lines produces two opposing effects: blending is less likely, but so is also the chance that mismatch errors will cancel statistically. But the advantage of fewer blends is largely negated when a large rotational velocity is present. The occurrence of broad and strong H and He lines, while offering possibilities for measuring RVs of rapidly rotating stars, presents other specific problems because of blending and spectrum rectification. In comparison with stars of later types, a disproportionate number of early-type stars exhibits spectral peculiarities that tend to be dominated by the behaviour of certain elements or ions (as in Bp and Ap stars). In stars of the earliest spectral types, some lines are formed in atmospheric velocity fields as large as several kms^-1(e.g. Ebbets [1979]). It will therefore be appreciated that RV measurements of early-type stars are prone to systematic errors that can amount to several kms^-1. Furthermore, they cannot be anchored to the late-type absolute zero-point in a simple way. There has therefore been little progress in defining a system of early-type RV standard stars (see Latham & Stefanik [1992]); the lack of early-type stars with RVs known accurately from spectrum-independent measurements is an additional drawback.

A reduction in error below the 1 kms^-1 level for the RVs of early-type stars is important for a variety of astrophysical studies, such as the kinematics and dynamics of young stellar groups, duplicity among early-type stars, and correction of high-precision proper motions for perspective acceleration. A general conclusion which we draw from previous studies (see Sect. 2.1) is that a significant improvement in the accuracy of relative RV measurements of different early-type stars, especially in the presence of rotation, may only be achieved by a more "microscopic'' approach of identifying and eliminating spectrum mismatch. It is evident that current efforts to establish a system of early-type RV standards to compute absolute RVs (Stefanik [1997]; Fekel [1999]) require the development of techniques that will specifically avoid spectral-type dependent relative errors (see Verschueren [1995]).

In this series of papers, we address questions such as: How large are the errors introduced by cross-correlating spectra of different types of stars between which different varieties of mismatch are likely to occur? Which spectral regions are more favourable than others? To what extent can accuracy be optimized by eliminating unreliable spectral regions? For a given wavelength region selected to obtain a sufficiently small systematic error, what S/N is needed to obtain a sufficiently small random error?

In this first paper we seek a better understanding of the relation between spectrum mismatch caused solely by given differences in $T_{\rm eff}$ and logg (hereafter referred to as "spectral-type mismatch''), and the consequent RV error (hereafter referred to as "mismatch shift''). This type of mismatch is all pervasive and is probably the main source of systematic errors. We conduct the experiments with synthetic spectra in order to isolate the temperature/gravity mismatch from additional sources of mismatch not reproduced by the models. The latter will be handled separately, on the basis of purely observational spectra, in subsequent papers. Further advantages of the use of synthetic spectra are given in Sect. 2.3. In this paper, we restrict the investigation to A-type stars on the main-sequence, providing a smooth connection to RV measurements of late-type stars.

Section 2 summarizes previous work, gives an example of the problems we are dealing with and outlines the methodology of our approach. Section 3 describes in detail our experimental set-up and the parameter-space covered. The mismatch shifts arising from small individual spectral regions are presented and discussed in Sect. 4. Section 5 proposes possible strategies for the selection and combination of spectral regions to minimize systematic errors, taking also into account the effects of S/N which are inevitable in observed spectra. Section 6 lists our conclusions and proposes future work.