2 Methodology

2.1 Previous work

The literature describes several systematic approaches for improving the accuracy of early-type RV measurements. Andersen & Nordström ([1983]) examined the suitability of individual spectral features in rotating late B- to F-type stars, basing their study on photographic spectra at 20 Åmm^-1. Measurements of individual lines indicated that many lines are affected by errors of up to 10 kms^-1, depending on spectral-type and rotation. The rms error of a single measurement of any one of their most reliable lines is 3 - 5 kms^-1, averaged over all stars, and a set of effective wavelengths was proposed for those lines, appropriate for the ranges of temperature and rotational velocity considered. Their results cannot easily be generalised or compared to ours because of the dependence of those wavelengths on resolution, because of the very different sensitivity to spectrum mismatch of their line-centering technique compared to cross-correlation, and because the wavelength regions we tested (Sect. 3.4) always contain much more spectral information.

A scheme to reduce the effects of mismatch in cross-correlations, by incorporating a set of observed early-type template spectra of different spectral types (and small rotational velocity), was developed systematically by Liu et al. ([1989]). A problem in using observed templates is of course that the stars selected may be RV variables, or that their actual velocities may not be known with sufficient accuracy. Liu et al. selected 19 $\rm B2 - A2$stars with presumed constant RVs and determined their relative RVs by mutual cross-correlation, using a region 150 Å wide around H$\delta$; more weight was given to results derived from adjacent spectral-types. That template grid was then tied to the late-type absolute zero-point in one of two ways: by cross-correlating the coolest member of their early-type set with a late-type RV standard, or by demanding that the mean of all velocities agree with the mean of all published velocities for those stars. The accuracy of their system seems to be $\sim$ 2 kms^-1, as judged from the difference in zero-point yielded by their two absolute calibration methods, from their observations of early-type stars in the Pleiades (Liu et al. [1991a], [1991b]), and from independent measurements of the latter stars by Morse et al. ([1991]). Owing to a sliding of the zero-point through the spectral sequence, errors are probably dependent on spectral-type. We argue below that a significant improvement in accuracy may be obtained by using a denser grid of templates, and by cross-correlating selected spectral regions. Morse et al. ([1991]) were the first to introduce the concept of a dense synthetic template grid of temperature and rotational velocity. Their models were based on Kurucz ([1979]) model atmospheres, from which they selected 2 spectral regions that are dominated by Balmer lines. Zero-point sliding through the grid, caused by the varying spectrum mismatch between synthetic and real spectra, was minimised by quite strong low-frequency fourier filtering of the spectra prior to cross-correlation. The degree of filtering was determined by inspection of the cross-correlation peak, and by demanding that their derived RVs of a set of frequently and carefully observed stars in the range early B to early A agree with their published values (mostly from Fekel [1985]) to within $\sim$ 1 kms^-1. By applying their method to observations of mid B- to late A-type stars in the Pleiades and $\alpha$ Persei clusters, Morse et al. demonstrated that the early-type zero-point defined by their template/filter grid agreed on average with the late-type absolute zero-point to within $\sim$ 1 kms^-1. Although the method was judged successful, anchoring a template grid to a consistent zero-point by the "right'' amount of filtering is nevertheless a casual result rather than a purposeful elimination of particular mismatches (see e.g. Verschueren [1991] for experiments with low-frequency filtering to handle mismatch between spectra). A more physically justified approach may lead to an even better and more robust accuracy.

Although it does not deal with early-type stars, the study by Nordström et al. ([1994]) should be mentioned in this context since it incorporated closely matching synthetic templates for rotating F-type stars. Using the wavelength region 5166 - 5211 Å, they found errors of up to 3 kms^-1 for vsini $\leq$ 100 kms^-1, although it is not clear how much was due to systematic errors and how much to random ones. They further found zero-point offsets of up to $\pm$ 1 km s^-1between their results and those from CORAVEL for a variety of rotational velocities up to 50 kms^-1.

Finally, the most successful (and ongoing) project in this field is that of Fekel ([1985], [1999]) who is monitoring several tens of candidate RV standard stars in the range B2-F2 in order to eliminate those with variable RV. By using relatively isolated MgII, FeII and TiII lines in the wavelength region 4460- 4550 Å, and by selecting stars with vsini < 50 kms^-1, Fekel is able to employ only 2 template spectra, the absolute RVs of which were determined by cross-correlation with late F-type standards. Morse et al. ([1991]) provide indirect evidence that Fekel's RVs are consistent with the late-type zero-point to within $\sim$ 1 kms^-1 without spectral-type dependence. Using the most recent velocities of Fekel ([1999]), we recomputed the average difference and rms spread between Morse et al. ([1991]) and Fekel ([1999]) for the 10 non-variable stars common to both studies and obtained 0.3 $\pm$ 0.9 kms^-1, thus adding firmness to the earlier conclusion. In Sect. 4.4 we confirm the appropriateness of that wavelength region for the stars studied by Fekel, and indicate how the situation changes for higher rotational velocities.

2.2 A closer look at mismatch

Figure 1: Top panel: wavelength region of the synthetic spectrum discussed in the text with $T_{\rm eff}$ = 7500 K at vsini = 5 (left) and 150 kms-1 (right). Middle panel: same for $T_{\rm eff}$ = 8000 K. The bottom panel shows the cross-correlation peak in both cases, centred at +0.01 kms-1 for vsini = 5 kms-1 and at -12.3 kms-1 for vsini = 150 kms-1 (in the sense: 7500 K spectrum with respect to 8000 K spectrum)

The following example illustrates the problems we are confronting here. Consider two noise-free synthetic spectra generated with the same rotational velocity (150  kms^-1), abundances (solar) and surface gravity (logg = 4.0), but with $T_{\rm eff}$ = 8000 K (spectral-type $\sim$ A7V) and 7500 K (spectral-type $\sim$ A9V), respectively. The spectra were degraded to a resolution of $\sim$ 0.12 Å per pixel at 4300 Å, and we select a wavelength interval 3717.1 - 5158.0 Å. Obviously there should be no Doppler shift between those spectra. Yet if we cross-correlate them, we find that the maximum of the cross-correlation funtion is at 2.2 kms^-1 instead of zero. If we then take smaller sub-regions, 4684.9 - 4724.9 Å and 5045.7 - 5060.9 Å, the mismatch shifts are 0.1 kms^-1 and 12.3 kms^-1, respectively. So the mismatch shift is by no means uniquely related to the difference in the atmospheric parameters only. A more detailed inspection shows that its magnitude and even its sign depend in the first place on the wavelength interval used in the cross-correlation, or rather on the behaviour and the importance of the blends therein; any blend which contributes significantly to the cross-correlation function and whose asymmetry is sensitive to $T_{\rm eff}$ and logg, may cause serious trouble in this context. For the last wavelength region considered above, Fig. 1 illustrates how differences, between object and template spectrum, in the relative strengths of nearby lines are harmless as long as lines are symmetrical and isolated - as is statistically the case at low rotational velocities. However, the fact that the relative strength of the FeI and NiI lines shortward of 5052 Å with respect to the CI line just longward of 5052 Å is significantly larger at 7500 K than it is at 8000 K places more weight, for higher rotational velocities, on the short-wavelength side of the blend at the lower temperature, causing a (large) displacement of the cross-correlation maximum.

A limited quantitative analysis of the connection between mismatch, caused by simple but specific differences in blending between object and template spectrum, and the resulting mismatch shift was presented by Verschueren ([1991]). He conducted cross-correlation experiments with a single, well-sampled, synthetic Gaussian line as template, and an object spectrum consisting of the same line blended with a weaker Gaussian line of the same width. Using the width $\sigma$ of the lines as the unit of length (the set-up is scale-invariant), and varying the distance d between the two line-centres in the object spectrum and the line-strengths of the primary (I₁) and blending (I₂) line, he concluded the following:

• the mismatch shift vanishes at d=0 and $d > \, \, \sim 6\sigma$; it reaches a maximum somewhere in the interval $\sigma < d < 2\sigma$;
• for a given I₁, the mismatch shift is proportional to I₂; for a given I₂/I₁, it depends only weakly on I₁;
• using a wavelength interval containing n similar blends (isolated from each other) and m isolated lines similar to the primary line in the blends, then the mismatch shift is approximately proportional to n/(n+m).

Since these results were obtained for an extremely simplified situation, we can only expect to be able to make qualitative comparisons with results from stellar spectra. In Sect. 4.3, we will see that this can be done to some extent when studying the influence of increased rotational broadening.

2.3 Proposed methodology

A straightforward approach to optimize the accuracy of RV measurements might seem to be to hunt down all "badly behaved'' blends (i.e. those sensitive to small changes in temperature or gravity) and somehow eliminate them from the cross-correlation process. However, one may then end up with so little "good'' material left that random errors become very high; this is especially true for fast rotators, partly because of increased blending and also because their cross-correlation peak positions are more sensitive to noise. The selection process must therefore take into account the effects of noise in the data, and seek a satisfactory compromise between random errors due to noise and systematic errors due to mismatch. Furthermore, eliminating a blend almost inevitably introduces into the data two artificial discontinuities, which may cause a serious systematic error (henceforth referred to as "end effect''). One may therefore be obliged to include wider spans, tolerating some bad blends and sacrificing some good regions.

At this point one could choose an ab initio approach to search for harmful blends by comparing line-lists of all elements contributing to the spectra and deducing, from the way their strengths vary with $T_{\rm eff}$ and logg, the behaviour of the blends according to a particular broadening law. Instead, we preferred a more phenomenological approach, since it offers directly a link with the magnitude of the induced mismatch shift:

• generate synthetic stellar spectra over a fairly wide wavelength range for fixed (solar) abundances, forming a grid of $T_{\rm eff}$ and logg plus a range in vsini;
• divide the whole wavelength range into a number of intervals subject to the requirements outlined in Sect. 3.4;
• for each interval, calculate the mismatch shifts which occur when cross-correlating it with the corresponding intervals in a set of neighbouring spectra in the ( $T_{\rm eff}$, logg)-grid.

The use of synthetic spectra for this purpose offers a number of advantages: an arbitrarily dense grid in $T_{\rm eff}$ and logg, perfect control over the parametrization, a large wavelength range, RVs that are exactly zero, and absence of noise. Even though synthetic spectra cannot be expected to be fully realistic in an absolute sense, their use here is justified because we only employ them differentially. Thus we anticipate that the synthetic spectral mismatch caused by a small difference in atmospheric parameters is in fact sufficiently realistic to reproduce the same kind of mismatch shifts one would find between real spectra, thereby providing us with the insight we are seeking. Obviously, any definite conclusions and a fortiori any tentative strategy, emerging from such a study, to reduce the errors must be verified using real spectra.

We shall not include any mismatch arising from abundance differences between object and template spectrum, as the parameter-space would become unmanageable; incidentally, that assumption may be quite appropriate for cross-correlating different stars from the same stellar cluster (e.g. Vrancken et al. [1997]). Abundance differences can be included at a later stage when studying individual stellar peculiarities in real spectra, as explained in Sect. 6. We also shall not consider mismatch arising through differences in rotational velocity because that can be avoided up to almost any accuracy by applying artifical rotational broadening to the template, and because - in the absence of other mismatch and of low-frequency filtering - it produces negligible systematic errors (Verschueren [1991]).