6 Conclusions and future work

The measurement of accurate RVs of early-type (O-B-A) stars cannot be achieved with the standard cross-correlation techniques that are suitable for most late-type stars, where object-template spectrum mismatch and selection of wavelength range are not critical points. For early-type stars the occurrence of spectral-type mismatch commonly results in large systematic RV errors ("mismatch shifts'') because their typically high rotational velocities cause spectral lines to blend into features that change shape with temperature and gravity, while their low line density prevents those errors from cancelling statistically.

In this first paper, we have used synthetic spectra in the range 3700 - 5200 Å and with a resolution of $\sim$ 0.12 Å per pixel to quantify the mismatch shifts that arise when differences between object and template spectrum of up to $\pm$ 500 K in $T_{\rm eff}$ ($\sim$ $\pm$ 2 temperature sub-classes) and up to $\pm$ 0.2 dex in logg ($\sim$ $\pm$ 1 luminosity class) are considered, representing 14 cases of spectral-type mismatch; variations in vsini were not included as a source of mismatch. The mismatch shifts that were derived from our cross-correlations are valid only for spectra in which spectral features are well sampled. The expected mismatch error $E_{\rm RV}$, conservatively defined as the maximum mismatch shift that can occur for all 14 mismatch cases, was derived for each of 30 main-grid spectra whose range spanned the rotating A-type main-sequence stars ( $T_{\rm eff}$ = 7500, 8000, 8500, 9000 and 9500 K, logg = 4.0, and vsini = 5, 50, 100, 150, 200 and 300 kms^-1). We derived mismatch errors for a set of several tens of spectral regions, five including H lines, in each spectrum.

We derive the following conclusions from our analyses of the mismatch errors of individual spectral regions in different spectra.

• A large range in mismatch errors arises between the different spectral regions for any given spectrum: the 10% highest errors are on average about a factor 20 larger than the 10% smallest ones;
• For the metal-line regions, the sign of the mismatch error is not randomly distributed in different spectral regions for many of the spectra with vsini $\geq$ 100 kms^-1;
• Mismatch errors tend to increase monotonically with vsinibecause blending becomes more severe with increasing rotational blurring. On average, mismatch errors scale linearly with vsini, but there is a large scatter owing to different degrees of individual blending;
• For a given spectral region, the mismatch errors may vary up to a factor 10 across the A-type temperature sequence. For most of the spectral regions, the variation of $E_{\rm RV}$ with temperature is monotonic but varies substantially in sign and in range from one region to the next, according to individual details of the spectrum mismatch. Average mismatch errors for the "best'' regions, i.e. the ones with a weak temperature dependence and a small average $E_{\rm RV}$, are $\sim$ 0.04, 0.4, 1, 1.5, 2 and 3 kms^-1 for vsini = 5, 50, 100, 150, 200 and 300 kms^-1, respectively;
• Mismatch errors from H-line regions lie well within the same range as those from metal-line regions. Errors arising from those containing H$\delta$, H$\gamma$ and H$\beta$ are relatively small for all spectra. The region containing H$\epsilon$ yields relatively large mismatch errors in mid- and late-A spectra owing to blending with the CaII H and K lines. The region containing H14 - H8 yields worsening errors for late A-type spectra when vsini $\geq$ 100 kms^-1.

Because statistical cancelling of errors is very inefficient, the mismatch error may increase substantially when more spectral regions are included in the cross-correlation. Accidental cancelling of large errors among individual spectral regions may also occur but will depend on spectral details that are not predictable in real spectra of individual stars. The development of a robust strategy for selecting individual spectral regions in order to minimize the mismatch error of the resulting combined region is therefore mandatory.

In view of the above considerations, we selected the different sets of spectral regions given in Table 2 for the 5 different ranges in vsini; they are suitable for the whole of the A-type main-sequence in each case. We made separate selections for the metal-line and for H-line regions. The use of those selections meets a number of important requirements: accidental cancelling of large errors is avoided, sufficient spectral information is included, and the selection is uniform in $T_{\rm eff}$ and continuous in vsini. For vsini $\leq$ 100 kms^-1, the selection based on metal-line regions leads to smaller errors than that based on H-line regions. For vsini > 100 kms^-1, regions with H lines may be superior; the latter depends in practice on the additional mismatch arising from differences in spectrum rectification, a step which is especially cumbersome for echelle spectra (e.g. Verschueren et al. [1997] and references therein). Our wavelength selection yields expected maximum mismatch errors for all A-type main-sequence spectra of $\sim$ 0.05, 0.15, 0.5, 1, 1.5 and 1.7 kms^-1, respectively, for vsini = 5, 50, 100, 150, 200 and 300 kms^-1. To ensure random errors not exceeding those systematic errors, a S/N of approximately 50 - 400 is required, depending on individual cases.

The size of the mismatch errors quoted above depends strongly on the size of the spectrum mismatch that we considered probable. If a half step in $T_{\rm eff}$ has to be bridged, the mismatch errors will be approximately halved owing to the quasi-linear dependence of the mismatch shifts on $T_{\rm eff}$ over that range. Decreasing the step in logg only becomes important if the mismatch error is no longer dominated by differences in $T_{\rm eff}$. Detailed spectrum matching, using a dense grid of template spectra, therefore remains of key importance for increasing the RV accuracy. Expected errors may also be reduced by relaxing the requirement that one set of selected spectral regions must be suitable for the whole of the A-type temperature sequence. That will be a natural refinement as the present study is extended to include B- and O-type spectra.

We stress that the selection of spectral regions proposed here is based solely on a phenomenological study of synthetic spectra, and must therefore be regarded as tentative and subject to verification from studies with observed spectra. We already showed in Sect. 4 that this study confirms the suitability of the wavelength region used by Fekel ([1985], [1999]) for all A-type stars with vsini < 50 kms^-1, but at the same time hints at problems that will arise for faster rotators. The next paper, which will study real spectra of A-type stars, should enable us to gain insight into the possible breakdown of some of our conclusions under conditions of other types of mismatch, such as arise (for example) through differences in chemical abundances. By using real spectra, we will be forced to confront the complex inter-relationships between the many atmospheric parameters which the use of synthetic spectra has allowed us to handle as independent and controllable quantities, and to investigate the less well understood links between those parameters and others arising from other atmospheric variants not included in the model atmospheres. Future work will also investigate the identification of those spectral features that are chiefly responsible for large mismatch errors (and for the non-randomness of their sign). By combining those results with the technical experience gained in the present study, we hope eventually to be able to propose wavelength-selections that are firmly based on astrophysical arguments and which are likely to be workable in practice.

Acknowledgements

We are very grateful to Herman Hensberge for many enlightening discussions in the course of this project and for a critical reading of the manuscript. Myriam Vrancken is thanked for valuable help with the synthesis of the spectra. WV acknowledges financial support from the Fund for Scientific Research - Flanders (Belgium) (F.W.O.) through Research Grant No. 1.5.549.98. REMG is grateful to RUCA for the support of a Visiting Professorship.