3 Experimental set-up

3.1 Synthetic spectra

Classical LTE line-blanketed atmosphere structures, having a solar composition and $v_{\rm turb}$ = 2 kms^-1, were taken from the grid of Kurucz ([1979], [1993]). Corresponding LTE spectra were computed with the programme SYNSPEC (Hubeny et al. [1994]), which takes Kurucz line lists (Kurucz [1988]; Kurucz & Bell [1995]) as input. Metal-line broadening was computed from classical expressions given by Kurucz ([1979]), while tables given by Vidal et al. ([1973]), Barnard et al. ([1974]) and Shamey ([1969]) were used for H and HeI.

The spectra were computed from the Balmer jump to 5200 Å and are sampled with a constant step of $28\, 10^{-6}$ in $\ln\lambda$(corresponding to a resolution of $\sim$ 0.12 Å per pixel at 4300 Å). This step-size provides that all apparent features in spectra broadened by $\sim$ 15 kms^-1 or more are well sampled. Since line blending is the cause of the errors we are studying, our results will not be strictly applicable quantitatively to spectra in which the same blends are not well sampled owing to inferior resolution (depending on vsini). This is in principle the case for our own synthetic spectra with vsini = 5 kms^-1: the mismatch shifts we compute from them will be somewhat larger than could ideally be achieved with greater resolution.

Atmospheric parameters were selected in the intervals $T_{\rm eff}$ = 7000-10000 K in steps of 250 K, and log g = 3.8 -4.2 in steps of 0.2, corresponding roughly to the range early F - late B sub-giants, dwarfs, and ZAMS stars. The spectra were rotationally broadened by vsini = 5, 50, 100, 150, 200 and 300 kms^-1. The largest rotational velocity was included only for a qualitative comparison with vsini = 200 kms^-1, since classical rotational broadening becomes a poor approximation near the break-up velocity of a star (see e.g. Collins & Truax [1995]). All spectra were normalized to a pseudo-continuum level by a smooth function fitted to the flux maxima between the Balmer lines of the spectra in which vsini = 5 kms^-1.

3.2 Spectrum mismatch cases

This phenomenological study has to simulate the spectral-type mismatch that is likely to occur between two main-sequence A-type stars. When comparing an observed, non-abnormal spectrum with the templates from a synthetic grid, the uncertainty in matching is probably not more than one sub-class in $T_{\rm eff}$ ($\sim$ 250 K) and about half a class in logg (0.1 dex). Thus, if we compute relative RVs between different observed stars whose classifications are each uncertain by that amount, the spectral-type mismatch may be doubled. Also, when establishing the relative RVs of candidate standard stars in a grid whose separations are one sub-class in temperature and half a class in gravity, we must attain high accuracy when bridging at least two such separations in order to ensure zero-point consistency throughout the whole system. For this study, we therefore investigate spectral-type mismatch produced by differences of up to two sub-classes in temperature ($\pm$ 500 K in $T_{\rm eff}$) and up to one class in gravity ($\pm$ 0.2 dex in logg).

In practice, that concept is translated into a "mismatch grid'' of 14 spectra, centred on a spectrum with given $T_{\rm eff}$, loggand vsini and differing from it by one and two sub-classes in $T_{\rm eff}$ and by one class in logg in either direction, respectively. Those 14 spectra represent 14 possible "mismatch cases'' to be considered for the central spectrum. Note that the difference of 0.2 dex we consider in logg was not subdivided further since mismatch shifts arising from this difference are already much smaller in general than shifts arising from the selected temperature mismatch. For the reasons given in Sect. 2.3, we disallowed the possibility of mismatch in vsini.

3.3 Mismatch shifts and mismatch errors

In order to study how spectral-type mismatch errors behave throughout the A-type main-sequence, we consider a "main grid'' of 30 spectra having $T_{\rm eff}$ = 7500, 8000, 8500, 9000 and 9500 K, logg = 4.0, and vsini = 5, 50, 100, 150, 200 and 300 kms^-1. These spectra may be regarded as templates. Each was then cross-correlated with the 14 spectra of its mismatch grid defined in Sect. 3.2; the latter spectra may be regarded as 14 object spectra which, because of our assumed classification uncertainties, may all be cross-correlated with the same template. All cross-correlations are carried out using the different spectral regions appropriate for each of the main-grid spectra (see Sect. 3.4). The result of each cross-correlation is a mismatch shift, defined as the difference in velocity caused by the mismatch between both spectra in the spectral region considered.

Next, for each of the 30 main-grid spectra, and for each spectral region, we define the expected "mismatch error'' $E_{\rm RV}$ as the maximum of the absolute values of the mismatch shifts derived for the 14 different mismatch cases. Since in practice one does not know which of the 14 cases occurs when cross-correlating two observed spectra, and since mismatch shifts are purely systematic in nature, the only useful parameter for characterizing the quality of a spectral region is the maximum value of all possible errors.

In summary, we have selected 30 main-grid spectra representing the A-type main-sequence stars at different rotational velocities; for each one we stipulated 14 probable cases of spectral-type mismatch and computed for each one the mismatch shift in velocity for a few tens of spectral regions. We defined, for each of the main-grid spectra, the mismatch error of each spectral region to be the highest of those shifts.

Cross-correlations were performed with the CORSPEC package, updated from Verschueren ([1991]). In order to minimize end effects, the flux in each spectral region was first rescaled to the average level between its two end points; an end-masking of 5 pixels length was then additionally applied. Mismatch shifts were computed by fitting a parabola through the highest 3 pixels of the cross-correlation peak, taking into account the discretization correction of David & Verschueren ([1995]). The formal random error for each cross-correlation was also computed using the theoretical expression given by Verschueren & David ([1999]) and adopting reference S/N values of 50, 100 and 200, respectively, in the continuum of the object spectrum and infinity for the template. The S/N in the continuum is assumed to be independent of wavelength, although in practice it depends on the intrinsic energy distribution of the spectrum and on interstellar, telluric and instrumental characteristics.

3.4 Spectral regions

Since we want to examine the behaviour of different parts of the spectra, we have to define a set intervals, which we call "spectral regions'', for the main-grid of 30 spectra defined in Sect. 3.3. Owing to the large differences in character between the different spectra, it proved impossible to select a unique set of spectral regions that was equally suitable in all cases. Nevertheless, we tried to define them as consistently and as homogeneously as possible for all the spectra considered, making neither their length too large that individuality was lost, nor so small that we ran the risk of low information content and important end effects.

   

Table 1: Master list of central wavelengths of continuum windows. Those windows bracketing a H line are indicated by that line in the last column. Continuum windows 01 and 02 bracket the Balmer series $\rm H14 - H8$. The start and end wavelength of all spectral regions for all spectra are chosen from this list. Throughout this paper, spectral regions will be noted by the combination of their numbers as appearing in the first column (e.g. region No. 08-12 runs from 4212.5 - 4405.5 Å)

No.	$\lambda$ (Å)	ln($\lambda$)		No.	$\lambda$ (Å)	ln($\lambda$)
01	3717.1	8.2207		19	4608.7	8.4357
02	3924.5	8.2750	H$\epsilon$	20	4623.9	8.4390
03	4019.8	8.2990	H$\epsilon$	21	4641.5	8.4428
04	4039.2	8.3038		22	4661.1	8.4470
05	4050.1	8.3065	H$\delta$	23	4688.2	8.4528
06	4159.7	8.3332	H$\delta$	24	4725.8	8.4608
07	4193.1	8.3412		25	4750.5	8.4660
08	4212.5	8.3458		26	4778.1	8.4718	H$\beta$
09	4231.4	8.3503		27	4944.3	8.5060	H$\beta$
10	4257.3	8.3564		28	4974.1	8.5120
11	4266.3	8.3585	H$\gamma$	29	4995.0	8.5162
12	4405.5	8.3906	H$\gamma$	30	5026.1	8.5224
13	4439.1	8.3982		31	5045.7	8.5263
14	4478.3	8.4070		32	5061.9	8.5295
15	4486.4	8.4088		33	5092.4	8.5355
16	4511.6	8.4144		34	5115.9	8.5401
17	4537.8	8.4202		35	5158.0	8.5483
18	4569.2	8.4271		36	5177.6	8.5521

First, in all 30 main-grid spectra, we searched in an automated way for "continuum windows'', defined here as spectral intervals in which the normalised flux is greater than 0.99 over a minimum span of 10 pixels for all nearby spectra with which the given spectrum will have to be cross-correlated (i.e. the set of 14 mismatch cases to be considered; see Sect. 3.2). A length of 10 pixels provides enough scope to ensure sufficiently flat ends in the event of even somewhat larger mismatches, or to accommodate actual Doppler shifts between observed spectra. For higher rotational velocities, the value of 0.99 had to be relaxed to 0.98 or 0.97 in order to find a reasonable number of continuum windows. For the late A-type spectra, no continuum windows shortward of H$\delta$ were found because of the increasing metal-line density in the Balmer-line wings; we decided simply to use the ones suitable for the other spectra in this wavelength region, and verified that resulting end effects were negligible in these relatively wide H-line regions.

From these data, a "master list'' of 36 continuum windows was generated which provided a satisfactory representation of all 5 main-grid spectra at vsini = 5 kms^-1. Table 1 lists their central wavelengths. At higher rotational velocities, some of those 36 windows cease to exist above a certain value of vsini because of line broadening, the more so for greater line-densities (i.e. lower temperatures). We therefore created sub-lists of continuum windows for different values of vsini and $T_{\rm eff}$, keeping the number as high as possible and the selection as homogeneous as possible across all temperatures and rotational velocities. In that way we retained 13 continuum windows at vsini = 300 kms^-1 for $T_{\rm eff}$ = 7500 K, and 23 for $T_{\rm eff}$ = 9500 K.

"Spectral regions'' were then defined simply as the intervals between adjacent continuum windows taken from the relevant sub-list, the whole set thus covering the entire spectrum synthesized in each case. A final iteration of the continuum windows was then carried out in order to ensure robustness against end-effects: we checked that end-masking and the actual location (up to $\pm$ 5 pixels) of the extremities of the spectral regions was not critical for the cross-correlations. These tests resulted in the removal of only a few continuum windows from some sub-lists defined for vsini $\geq$ 100 kms^-1. Owing to the constraints we imposed on the continuum windows, some of the small spectral regions merge into larger ones as vsini increases and $T_{\rm eff}$ decreases. The final lists of spectral regions include, for all spectra, 5 "H-line regions'': the H14-H8 Balmer line region, one with H$\epsilon$ and CaII K, and those with H$\delta$, H$\gamma$ and H$\beta$ - each of course blended with other lines. Regions without H lines (the "metal-line regions'') have an average width of $\sim$ 25 Å if vsini = 5 kms^-1 and $\sim$ 40 Å if vsini = 300 kms^-1.