4 Rotational object-template mismatch

  

Figure 7: a) Random error on a cross-correlation shift as a function of the number of fit-points used (for the quadratic fit-function) as derived from a Monte-Carlo experiment with 1000 runs. A B1V synthetic spectrum in the region 4547-4759 Å was used. The object spectrum has a rotational velocity of 200 kms-1 and a S/N = 50. The rotational velocity of the template spectrum varies as indicated and its S/N was 200. The lower bound for the random error (Sect. 2), in absence of rotational mismatch, is 0.58 pixel. Note that the fit-interval is not normalised here to the FWHM of the cross-correlation peak since the latter changes with rotational mismatch (from 35 to 39 pixels). b) Idem as a) but now the rotational broadening of the template to 200 kms-1 was done after the noise was added

The question we raise here is how the random error on a cross-correlation derived radial velocity shift behaves when the template has a lower rotational velocity than the, otherwise intrinsically identical, object spectrum.

Monte-Carlo tests were performed similar to the ones described in Sect. 3: a sharp-lined synthetic spectrum was once broadened to a given value of vsini to yield the object spectrum, and once broadened to several lower vsini values to simulate different template spectra. Random noise was independently added to both and the shift was computed after cross-correlation. No Fourier filtering was applied in order to show the pure effect of rotational mismatch. As an example, Fig. 7a shows the rms of these shifts for different values of the rotational velocity of the template. The conclusion is that rotational mismatch does not increase the minimum random error obtainable in case the spectra are rotationally matching; this minimum error, however, can now be achieved in practice only for much larger fit-intervals, having lengths within a smaller range (in this case between $\sim45$ and 55 pixels). The reason is that the random noise on the cross-correlation function has increased due to the poorer line overlap.

The last curve of the experiment just described correponds to the case of using an observed template with a vsini matching the object as close as possible. Alternatively, one may also use a template consisting of an artificially rotationally broadened version of a sharp-lined observed spectrum. The important difference between the two cases is that in the latter, rotational broadening is done after addition of noise. Figure 7b shows the effect on the random error in the latter case: the minimum random error obtainable remains roughly unaltered, but it can now be achieved for the smallest number of fit-points possible. This result is, not surprisingly, very similar to the effect of high-frequency filtering described in Sect. 3.3 since rotational broadening obviously smooths the spectrum; clearly, high-frequency filtering of the spectra would turn the last curve of Fig. 7a into that of Fig. 7b.

We conclude that rotational mismatch should be avoided for technical reasons only, namely for its implication on the fitting process of the cross-correlation peak. The decision to use either observed rotationally matching templates in combination with high-frequency filtering, or artificially broadened sharp-lined templates, must be based on the following two factors: on the one hand the availability of an observed rotationally matching template spectrum, and on the other hand the adequacy of straightforward rotational broadening to mimic vsini (which becomes questionable for v close to the break-up velocity; see e.g. Collins & Truax 1995).