5 Selection and combination of spectral regions

In this section we experiment numerically with several criteria for excluding the badly-behaved spectral regions in order to find out, from a phenomenological basis, to what extent mismatch shifts can be reduced while at the same time keeping the random error contained. An optimal selection strategy will depend on the systematic error that can be tolerated and on the random error (S/N) that is achievable. In this paper, we assume that the S/N is not in principle the limiting factor. We can therefore concentrate on selection strategies that minimize the systematic mismatch errors in an absolute sense, and compute the S/N necessary to keep the random errors below those systematic ones.

We consider the set of pure metal-line regions separately from the set of 5 regions containing H lines. The latter regions have much wider wavelength spans and the two sets have different sensitivities to rotational broadening. In addition, the particular non-random behaviour of the sign of the mismatch shifts (see Sect. 4.1) is only apparent for metal-line regions.

5.1 Sequential combinations

Figure 7: Crosses and the connecting full line show, for the three spectra indicated on the right, the mismatch error $E_{\rm RV}$ for the sequentially-combined spectral regions defined in Sect. 5.1. Dotted lines show the random error of each combined region for a $(S/N)_{\rm obj}$ = 50, 100, 200 from top to bottom, respectively, and a $(S/N)_{\rm tem}$ = $\infty$. Results are shown separately for combinations without, and with, H lines. Just above the top of the graph is indicated, for each combined spectral region, the number of the last-added individual spectral region (the notation is described in Table 1). The number just below the top of the graph is the $E_{\rm RV}$ value of the latter expressed in kms-1

We know that, for typical late-type spectra with a high line density and a low rotational velocity, mismatch shifts statistically cancel to a very large extent; systematic errors are therefore small, and random errors can be minimized by combining as many spectral regions as possible. This led us to combine spectral regions in a particular way for each of the 30 main-grid spectra in turn. We commence by selecting that region giving the smallest mismatch error $E_{\rm RV}$ in each case, and subsequently adding further regions in order of increasing $E_{\rm RV}$. Each step, which includes one more "individual'' region than the previous step, constitutes a "combined region''; the last combined region is thus the combination of either all metal-line regions or all H-line regions. Figure 7a shows the mismatch errors $E_{\rm RV}$ arising from those combined regions for the case with $T_{\rm eff}$ = 7500 K, vsini = 50 kms^-1. Also indicated is the $E_{\rm RV}$ corresponding to the most recently-added individual region, together with the random error associated with each combined region for three adopted values for the S/N in the object spectrum. In this case of our coolest spectrum with a low rotation, which may be considered as the "late-type analogue'', mismatch shifts are effectively cancelled throughout the metal-line regions without increasing the errors above the level for the best individual region. For the H-line regions, however, the inclusion of the region containing H14 - H8 and the one containing H$\epsilon$strongly increases the error, which can be understood from Sect. 4.5.

The example given in Sect. 2.2 has already shown that statistical cancelling of errors fails in the case of higher temperatures or greater rotational velocities. Figure 7b (corresponding to $T_{\rm eff}$ = 9500 K, vsini = 50 kms^-1) and Fig. 7c (corresponding to $T_{\rm eff}$ = 8500 K, vsini = 200 kms^-1) do indeed exhibit radically different behaviour from Fig. 7a. Owing to the lower line density and (at higher rotational velocities) to the apparently non-random sign of the mismatch shifts of individual spectral regions (see Sect. 4.1), cancelling of errors becomes very inefficient, and the expected mismatch error may substantially increase when more individual spectral regions with successively higher mismatch errors are added. Some (large) errors may still cancel for certain combinations of spectral regions, but the effect is accidental rather than statistical. Unless one is sure that all details of a synthetic spectrum are realistic, one certainly cannot count on the same cancellation occurring for real spectra. Verschueren et al. ([1999]) investigated, in a particular case, the extent to which mismatch errors may be reduced by searching explicitly for those combinations of spectral regions that show the largest accidental cancelling. For the reasons just explained, the outcome of such an experiment cannot provide the basis for the choice of a suitable set of spectral regions, at least insofar as only synthetic spectra are considered. Our policy for combining spectral regions is therefore to avoid using individual spectral regions with relatively large mismatch errors.

However, since the curves in Figs. 7b,c do not rise monotonically, the practice of avoiding only those individual spectral regions with the largest $E_{\rm RV}$ does not necessarily reduce the $E_{\rm RV}$ of the resulting combined region. The only possible robust strategy is thus to use just those spectral regions with very small individual mismatch errors. The cancelling of errors is then reduced to an acceptable level, though we note that some amount of cancelling is always unavoidable, e.g. as occurs internally within each of the individual spectral regions.

Figure 8: a) Full lines: Expected RV accuracy attainable for different ranges in vsini when using the "uniform'' selections of metal-line wavelength regions as explained in Sect. 5.2. The relationship is valid for all A-type main-sequence spectra and for the spectral-type mismatch outlined in Sect. 3.2. For vsini = 200 - 300 kms-1, the selection which is tentatively proposed in Table 2 yields errors of about 5 kms-1and is not shown for clarity. Dashed line: Same for the "uniform'' selections of H-line regions. Dotted line: Same when using only the best individual spectral region as explained in Sect. 5.1. b) S/N value necessary to obtain a random error equal to the systematic error given by the corresponding type of curve in a)

For the purpose of later comparisons, it is instructive to investigate the order of magnitude of the mismatch error $E_{\rm RV}$ which results when this strategy is taken to its extreme, i.e. in each of the 30 main-grid spectra, we use only that individual region which gives the smallest $E_{\rm RV}$ (regardless of whether it is a metal- or H-line region). For any given vsini, these $E_{\rm RV}$ values show some scatter with $T_{\rm eff}$, but there is no trend. Their maxima over the 5 values of $T_{\rm eff}$are therefore a conservative measure of the maximal RV accuracy attainable at each vsini, at least when one does not want to depend on cancelling of errors. The dotted curve in Fig. 8a shows this accuracy: RV accuracies below 1 kms^-1 are only possible for rotational velocities below about 150 - 200 kms^-1, but even at vsini = 300 kms^-1 those errors are definitely below 2 kms^-1. Unfortunately, that result has mostly academic value only, for three reasons. First, since very small wavelength regions are used in the metal-line case, the required S/N necessary to obtain a random error of the same order of magnitude as the systematic error is in the region of 1000 and is thus out of reach for most observing programmes at present. Note that this problem is reversed for a H-line region, where systematic errors are commonly larger than random errors even at moderate S/N(as shown by the dotted curves in Figs. 7). Secondly, that result depends entirely on the realism of just a few spectral features in a synthetic spectrum. And thirdly, the very best spectral region is a different one for different spectra; there is no guarantee that the best regions in two adjacent spectra will yield equally small errors at intermediate values of $T_{\rm eff}$ or vsini.

5.2 Uniform combinations

In view of the foregoing discussion we implemented the following strategy for selecting individual spectral regions. First, for each value of vsini separately, we selected those individual regions whose derived mismatch error $E_{\rm RV}$ is small for all $T_{\rm eff}$ (see e.g. Fig. 5). Because of the need for a sufficient number of selected regions, we selected $E_{\rm RV}$ cut-off values for the 6 different vsini cases of 0.1, 0.7, 1.5, 2.5, 3 and 7 kms^-1, respectively. Even then, the number of suitable regions decreases strongly with vsini. In order to ensure continuity as a function of vsini, only those regions common to the selections for two consecutive vsini values were then retained for use between those two vsini values. This procedure was followed separately for metal- and H-line regions but with the same $E_{\rm RV}$ cut-off values. Table 2 lists the individual spectral regions finally selected in this way. We note parenthetically that an alternative strategy, of searching at a given $T_{\rm eff}$ for spectral regions with a small $E_{\rm RV}$for all vsini, was unsuccessful.

   

Table 2: Numbers of the individual spectral regions selected for the "uniform'' combinations for different ranges of vsini as discussed in Sect. 5.2. They are suitable for the whole A-type main-sequence. The notation is taken from Table 1. For the H-line regions, the exact definition of the regions may change with temperature and rotation because of the changing availability of continuum windows around the H lines (see Sect. 3.4)

vsini (kms -1)	spectral regions
005-050	15-16, 16-17, 17-18, 18-19, 19-20, 20-22, 23-24, 34-35
050-100	15-16*, 17-18, 18-19, 23-24
100-150	15-16*, 18-19, 23-24, 27-28
150-200	18-19, 23-24, 27-28, 28-29*
200-300	18-19*, 28-29*
		7500 K	8000 K	8500 K	9000 K	9500 K
005-050	H$\beta$ (005)	26-27	26-27	26-27	26-27	26-27
	H$\beta$ (050)	26-27	26-27	26-27	26-27	26-27
050-100	H$\beta$ (050)	26-27	26-27	26-27	26-27	26-27
	H$\beta$ (100)	25-27	25-27	25-27	25-27	25-27
100-150	H$\gamma$+H$\beta$ (100)	09-13, 25-27	11-13, 25-27	11-13, 25-27	11-12, 25-27	11-12, 25-27
	H$\gamma$+H$\beta$ (150)	09-13, 25-27	11-13, 25-27	11-13, 25-27	11-12, 25-27	11-12, 25-27
150-200	H$\delta$+H$\gamma$ (150)	04-08, 09-13	04-08, 11-13	04-06, 11-13	05-06, 11-12	05-06, 11-12
	H$\delta$+H$\gamma$ (200)	04-08, 08-16	04-08, 08-13	04-08, 11-13	04-06, 11-12	04-06, 11-12
200-300	H$\delta$+H$\gamma$ (200)	04-08, 08-16	04-08, 08-13	04-08, 11-13	04-06, 11-12	04-06, 11-12
	H$\delta$+H$\gamma$ (300)	04-08, 08-16	04-08, 08-16	04-08, 11-13	04-06, 11-12	04-06, 11-12
*Region not defined for $T_{\rm eff}$ = 7500 K.

Our procedure meets the four requirements following from the discussion in Sect. 5.1. First, accidental cancelling of large errors is avoided because only very good individual regions are used; secondly, enough spectral information is sampled so as to avoid the need for extremely high S/N (see Fig. 8b) and (thirdly), so as not to depend on the details of just a few spectral features in a synthetic spectrum; fourthly, the selection is uniform in $T_{\rm eff}$ and continuous in vsini, so the derived errors are valid for the whole of the parameter space.

This selection was then applied to all 30 main-grid spectra and the expected mismatch error $E_{\rm RV}$ was computed for the standard mismatch defined in Sect. 3.2. At vsini = 50, 100, 150 and 200 kms^-1, both relevant selections were applied separately; e.g. at vsini = 50 kms^-1, the selection suitable for 5 - 50 kms^-1 and the one suitable for 50 - 100 kms^-1 were both computed. For a given vsini, the same selection was of course applied for the 5 values of $T_{\rm eff}$. The mismatch errors $E_{\rm RV}$ were then plotted as a function of $T_{\rm eff}$ for each value of vsini and, again, no significant dependence on temperature was found. Therefore, for each vsini, the maximum value over the 5 values of $T_{\rm eff}$ was adopted as a conservative measure of the accuracy attainable with each of the proposed selections. Figure 8a shows this expected accuracy for each of the ranges in rotational velocity considered, and for metal- and H-line regions separately. For vsini below about 100 km s^-1, our selection based on metal-line regions produces consistently smaller errors than that based on H-line regions. For vsini > 100 kms^-1, regions with H lines are superior. For vsini $\leq$ 150 kms^-1, errors are consistently smaller than 1 kms^-1 ($\sim$ 0.05, 0.15, 0.5 and 1 kms^-1 for vsini = 5, 50, 100 and 150 kms^-1, respectively), while for higher rotation errors in the range 1 - 2 kms^-1 can be expected. We recall that these error estimates are conservative values in the sense that they are maxima of errors over all temperatures, which in their turn were based on the maximum mismatch shift of all 14 mismatch cases in $T_{\rm eff}$ and logg considered (see Sect. 3.2).

Figure 8b shows the S/N necessary for the random error to equal the systematic error of Fig. 8a for each of the proposed selections of spectral regions. For the metal-line selections with vsini $\leq$ 100 kms^-1, S/N values roughly between 100 - 400 are required, while much smaller values are sufficient if the H-line regions are selected when vsini > 100 kms^-1. The latter stems from the fact that on the one hand the intrinsic cross-correlation power in the H-lines decreases only slightly with rotational velocity, while on the other hand the systematic error increases much more.