4 Mismatch errors of individual spectral regions

In this section, we present the expected mismatch errors $E_{\rm RV}$ of the individual spectral regions. We recall that each is defined as the maximum mismatch shift (in absolute value) out of the 14 different mismatch cases (two steps of $\pm$ 250 K in $T_{\rm eff}$ and one step of $\pm$ 0.2 dex in logg; see Sect. 3.2). We discuss these results for the 30 main-grid spectra considered, i.e. for the 5 values of temperature, 1 of surface gravity, and 6 of rotational velocity, representing the rotating A-type main-sequence stars (see Sect. 3.3). First, in Sect. 4.1, we look in more detail at the individual mismatch shifts arising from the 14 different mismatch cases.

4.1 Mismatch cases and shifts

The step in ( $T_{\rm eff}$, logg) of the mismatch grid as chosen in Sect. 3.2 needs to be sufficiently small to sample properly the mismatch error $E_{\rm RV}$ as a function of $\Delta T_{\rm eff}$ and $\Delta {\rm log\,}g$. We are satisfied that the latter requirement has in fact been met because a simple four-parameter model ( $A \Delta T_{\rm eff} + B \Delta T_{\rm eff}^2 + C \Delta {\rm log}\,g + D \Delta T_{\rm eff} \Delta {\rm log}\,g$) fits the 14 mismatch shifts fairly well for most spectral regions in all main-grid spectra. We also found that the maximum mismatch shift, i.e. the mismatch error $E_{\rm RV}$, for $\Delta T_{\rm eff} = 500$ K (i.e. for 14 mismatch cases) is twice that for $\Delta T_{\rm eff} = 250$ K (i.e. for 8 mismatch cases), with an rms deviation less than 0.5 for all 30 main-grid spectra and for all spectral regions.

Figure 2: Percentage of individual metal-line regions with a positive mismatch shift for those mismatch cases with $\Delta T_{\rm eff} > 0$ (full line), and for those with $\Delta T_{\rm eff} < 0$ (dashed line). All mismatch cases in logg are taken together. Results are shown for each of the 30 spectra separately with the numerical labels referring to their $T_{\rm eff}$ (1=7500K, 2=8000K, 3=8500K, 4=9000K, 5=9500K). If $\Delta T_{\rm eff} > 0 (< 0)$, the mismatch shift is that of a hotter (cooler) object with respect to the template whose $T_{\rm eff}$is referred to by their labels

As pointed out in Sect. 1, if spectra have a high line-density one tends to assume that, for a given mismatch case, the mismatch shifts caused by individual blends in a given spectrum are like random numbers distributed symmetrically around zero. Visual inspection of our mismatch shifts (see e.g. Fig. 2) suggests that, on the contrary they have a preferential sign in different spectral regions for many of the spectra and mismatch cases. Although the wavelength regions defined in Sect. 3.4 do not isolate individual blends, those without hydrogen lines are sufficiently narrow and numerous to allow a limited verification of the assumption of randomness of the sign. For each of the 420 samples of mismatch shifts taken from all metal-line regions (for 14 mismatch cases in 30 spectra), we tested the hypothesis that the sign of the shifts in the sample is random. The results can be summarized as follows:

• at the 5% significance level the hypothesis cannot be rejected for any of the 30 spectra if the mismatch is caused only by differences in logg and not in $T_{\rm eff}$;
• likewise it cannot be rejected for any mismatch case if the template spectrum has vsini = 5 or 50 kms^-1;
• however, for all temperatures there are one or two vsini values and several mismatch cases for which the hypothesis must indeed be rejected at the 5% level, with a slight preponderance of cases with $\Delta T_{\rm eff} > 0$, $\Delta {\rm log\,}g$ = 0 and $T_{\rm eff}$ > 7500 K; the total number of these cases is 110, and in 42 out of them the hypothesis must be rejected even at the 1% level so it is quite unlikely that they could be ascribed to coincidence.

A characteristic of these trends is illustrated in Fig. 2, where a strong correlation appears between the sign of the temperature mismatch and the sign of the corresponding mismatch shift, for most metal-line regions and for rotational velocities above $\sim$ 50 kms^-1. The correlation exists for all temperatures, although the scatter increases towards the highest vsini because of the smaller number of regions defined. The reason for this common behaviour of most spectral regions has yet to be investigated by examining the origin of each shift. Positional relations between the lines of multiplets contributing to temperature-dependent blends in different regions presumably play a role. The important consequence is that, at least for faster rotators, a majority of the metal-line blends may cause a mismatch shift in the same sense so that one cannot be certain of reducing systematic errors simply by including more such blends in a cross-correlation. This will strongly influence the extent to which it is useful to combine individual spectral regions as is discussed in Sect. 5.1.

4.2 Distribution

For each of the 30 main-grid spectra we plotted the distribution of mismatch errors $E_{\rm RV}$ from the different spectral regions. A Kolmogorov-Smirnov test shows that, for a given rotational velocity, the hypothesis that the distributions for the 5 different temperatures are identical cannot be rejected, although the relatively small number of regions must influence that result. We can therefore combine the results for all $T_{\rm eff}$; Fig. 3 shows the distributions for the 6 values of vsini. It is immediately apparent that a large range in $E_{\rm RV}$exists among the different regions at any given vsini: the 10% highest errors are on average about a factor 20 larger than the 10% smallest ones.

Figure 3: Distribution of the mismatch errors $E_{\rm RV}$ from all spectral regions. For each value of vsini, the results from the 5 values of $T_{\rm eff}$are combined. Results for H-line regions are shown hashed

The most important characteristic of these distributions is the strong and consistent increase in the errors with vsini. In Sect. 4.3 we will quantify that dependence in more detail. Although the distributions of all $E_{\rm RV}$ for different temperatures are indistinguishable, the mismatch error of each individual region may vary significantly with $T_{\rm eff}$, as discussed in Sect. 4.4. The smallest mismatch errors from the H-line regions are of very much the same magnitude as the errors of the best metal-line regions, while at the same time H-line regions are never found among those regions that give rise to the highest errors. A more detailed discussion of the H-line regions is given in Sect. 4.5.

4.3 Dependence on rotation

As Fig. 3 shows, the global distribution of mismatch errors scales approximately linearly with vsini. In order to investigate this dependence more quantitatively for individual spectral regions, those regions were selected that contain exactly the same interval in wavelength for all values of vsini. This was repeated for the 5 values of $T_{\rm eff}$ separately in order to detect a possible further dependence on temperature.

Figure 4: Normalised mismatch error $E_{\rm RV}$ as a function of vsini, plotted on a double logarithmic scale. Different curves represent all spectral regions that contain exactly the same interval in wavelength over the full range of vsini for spectra with $T_{\rm eff}$ = 8000 K. Dotted lines represent H-line regions. The thick line depicts Eq. (1), its slope representing a linear scaling of the error between any two values of vsini

A linear relation between the mismatch error from a given spectral region and vsini, assuming the error is zero if vsini = 0, can conveniently be expressed on a logarithmic scale:

$$\log \left( \frac{E_{\rm RV}}{E^{(5)}_{\rm RV}} \right) \, = \, \log \left( \frac{v \sin i}{5} \right)$$ (1)

with $E^{(5)}_{\rm RV}$ the mismatch error at vsini = 5 kms^-1 to which the errors for other vsiniare normalised. Figure 4 shows the mismatch error for $T_{\rm eff}$ = 8000 K as a function of vsini for all 8 selected regions; it also depicts Eq. (1). The double logarithmic scale has the advantage that a linear dependence of the mismatch error on vsini between any two vsini values is translated into a connecting line with a gradient of unity, i.e. parallel to Eq. (1).

As the plots in Fig. 4 are not significantly different for different temperatures, we can draw the following conclusions from all of them:

• $E_{\rm RV}$ increases monotonically with vsini in almost all spectral regions, partly because in a majority of the wavelength regions all individual mismatch shifts actually increase monotonically, and partly because $E_{\rm RV}$, by its very definition, is likely to keep on growing even if some of the individual shifts do not. The behaviour of the individual mismatch shifts themselves (monotonic or not) can often be explained qualitatively in terms of the results of the simple two-line study by Verschueren ([1991]) (see Sect. 2.2), bearing in mind that the distance between spectral lines in units of their width decreases as rotational broadening increases. In particular, in some spectral regions where the mismatch error exhibits a maximum as a function of vsini, one finds that the condition for a maximum is indeed reached and passed at some vsini < 300 kms^-1 and that all individual mismatch shifts decrease in absolute magnitude.
• Almost all curves have at least one part where the increase is stronger than linear and at least one part where it is less strong than linear. Furthermore, for any step in vsini, different spectral regions exhibit a large range in slope around the linear value. Although rotational broadening increases $E_{\rm RV}$ on average, the details of the blending in each individual region determine the final outcome to a large extent.
• The vsini-dependence of the mismatch errors arising from H-line regions does not seem any different from that arising from the metal-line regions. The reason is that mismatch errors in H-line regions are caused solely by the blending metal lines (with some exception occurring in the Balmer line region; see Sect. 4.5).

4.4 Dependence on temperature

For a given vsini, the mismatch error $E_{\rm RV}$ for any given spectral region may vary considerably with the temperature parameter $T_{\rm eff}$ of the spectrum. This is shown for vsini = 50 kms^-1 in Fig. 5. The following conclusions can be drawn:

• The mismatch errors of the majority of the spectral regions vary monotononically with temperature, although the sign and the range of the variations varies substantially from one region to the next. Again, the details of the blending in each spectral region are the crucial factor;
• Spectral regions with a relatively weak dependence on temperature also have a relatively small average $E_{\rm RV}$. Average errors for these "best'' regions 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. Note that these "best'' regions are largely different ones for different vsini; this seriously hampers the selection of regions suitable for all spectra (see Sect. 5);
• Some spectral regions have mismatch errors that vary up to a factor of 10 across the temperature range considered. Such large variations, however, do not occur among H-line regions.

Figure 5: Mismatch error $E_{\rm RV}$ as a function of the different spectral regions for 5 different temperatures: 1 = 7500 K, 2 = 8000 K, 3 = 8500 K, 4 = 9000 K and 5 = 9500 K. The rotational velocity is 50 kms-1in all cases. Spectral regions are indicated by the number of the continuum window corresponding to their starting wavelength (see Table 1). Horizontal full lines indicate the length of a region if it is larger than the distance between two consecutive continuum windows in Table 1

Figure 5 supports a comparison with the work of Fekel ([1985], [1999]), who has derived accurate RVs for late B- and A-type stars that have vsini < 50 kms^-1. Fekel uses the MgII line at 4481 Å (situated in our region No. 14-15) and the 5 FeII and TiII lines between 4500 - 4525 Å (situated at the end of region No. 15-16 and at the beginning of No. 16-17). Our error in region No. 14-15 (MgII) remains small for vsini = 5 kms^-1 but it increases drastically with decreasing temperature for vsini = 50 kms^-1(see Fig. 5). For that value of the rotation the region is still reliable for early-A spectra, but yields unacceptably high errors for mid- and late-A ones owing to blending by lines which have a strong temperature dependence. Incidentally, the mid and late A-type stars in Fekel's study all have vsini < 15 kms^-1 (Fekel 1999, private communication) which helps to explain the relatively small errors he finds for those stars using the MgII line. Also for vsini > 50 kms^-1, we find that the MgII line yields relatively small errors for early A-type spectra only.

The other two regions which contain lines used by Fekel are in fact among the very best, producing mismatch errors between 0.02 - 0.04 kms^-1 for vsini = 5 kms^-1 and between 0.2 - 0.4 kms^-1for vsini = 50 kms^-1 (depending on temperature). However, since the maximum mismatch in temperature we have considered spans only 2 temperature sub-classes in either direction, our results indicate that allowing a considerably larger mismatch, such as Fekel ([1985], [1999]) employs, is not without risk for vsini approaching 50 kms^-1. For vsini > 50 kms^-1, our region No. 15-16 remains among the best despite the blending which occurs shortward of Fekel's lines. At that rotational velocity, region No. 16-17 behaves well only in early A-type spectra, and produces relatively high mismatch errors for cooler spectra owing to a complex blend which affects most of Fekel's lines.

4.5 A closer look at H-line regions

Owing to the much larger widths and strengths of the Hydrogen lines compared to the metal lines in these early-type spectra, spectral regions that contain H lines require separate investigation. It is important to realize at the outset that the mismatch errors of the 4 regions that contain just one H line differ from zero only because of the $T_{\rm eff}$- and logg-dependent blending with metal lines; the H line itself is always isolated and symmetric. On the one hand, mismatch errors arising from those 4 H-line regions are expected to be smaller on average than those arising from regions with only metal lines because the strong, broad H line dampens extraneous contributions while not itself contributing to the mismatch; smaller errors for H-line regions are expected also because they have a much larger wavelength span than metal-line regions and because some degree of cancelling of errors always occurs (see Sect. 5.1). On the other hand, some differences in the characteristics of the metal lines (e.g. a difference in strength of an isolated line) will yield (larger) mismatch shifts if those lines occur on the inclined wings of a H-line profile; in addition, differences in spectrum rectification may be an important source of mismatch for these regions. The nett effect cannot easily be predicted, though the distributions given in Sect. 4.2 already showed that mismatch errors from H-line regions lie well within the range found for metal-line regions.

Figure 6: Mismatch errors $E_{\rm RV}$ of the 5 H-line regions as a function of spectral temperature for a rotational velocity of 150 kms-1. The regions are: 1 = H14 - H8 (Balmer region), 2 = H$\epsilon$, 3 = H$\delta$, 4 = H$\gamma$, 5 = H$\beta$. Dashed lines indicate a change in length of the actual wavelength interval, which occurs when a small adjacent metal-line region merges with the H-line region (see Sect. 3.4)

Figure 6 shows the mismatch errors $E_{\rm RV}$ of the 5 H-line regions for the 5 temperatures considered, and for vsini= 150 kms^-1. From this and similar plots for different vsiniwe conclude:

• Mismatch errors arising from the regions containing H$\delta$, H$\gamma$ and H$\beta$ (labelled 3, 4 and 5) are relatively small; they show no common dependence on temperature for any vsini, and in that sense they behave similarly to the metal-line regions. Since it is the detailed content of the metal lines in the H-line regions that determines the mismatch shifts, the addition of adjacent small wavelength regions to a H-line region (see the dashed lines) can have a non-negligible effect;
• For vsini < 50 kms^-1, the situation for region 2 containing H$\epsilon$ is similar to that of the 3 regions discussed above. For larger rotational velocities, however, mismatch is dominated by the blending with the CaII H line, though in mid- or late-A- spectra the K line also blends with the far wing of H$\epsilon$; both CaII lines vary markedly with temperature, growing from a very weak line at late B to strong and broad lines at early F. The resulting mismatch errors show a very strong temperature dependence and are relatively high for mid- and late-A spectra. This particular type of blending occurs only in the H$\epsilon$spectral region;
• For small vsini, the situation for region 1 containing the Balmer-series limit (H14 - H8) is again similar to that of the other H-line regions. For larger rotational velocities, the errors remain small for early- and mid-A spectra but become relatively large for late-A ones. We suggest the following tentative explanation. At larger vsini the mismatch is dominated by the seven mutually blended H lines; those lines become broader with increasing temperature, reaching a maximum at early A. Their blending therefore alters substantially with temperature for the late A-type spectra, while reaching a more stable, almost temperature-independent, configuration for the early A-type spectra. If that mechanism is indeed the dominant one, and if other factors remained equal, we might expect mismatch errors for that spectral region to increase again for B-type stars, where the H lines become more narrow with increasing temperature.