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3. Determination of rotational velocities

There are several factors to be considered when a spectral line is selected for measuring rotation velocities.

Very few lines fulfil these conditions in the La Palma spectra, mainly because of their moderate resolution which makes most of the lines appear blended. After a careful selection, we have chosen two lines: tex2html_wrap_inline2401 and tex2html_wrap_inline2403. The first one is stronger but it appears slightly blended with the weak line tex2html_wrap_inline2405 and it will be used only for tex2html_wrap_inline2407 tex2html_wrap_inline2409 tex2html_wrap_inline2411 where it is not possible to use the weaker line tex2html_wrap_inline2413. On the contrary, this problem is not present in the McDonald spectra where the spectral resolution is a factor of six higher. For these spectra, a set of nine lines was used in the calculation of the rotational velocities: tex2html_wrap_inline2415, tex2html_wrap_inline2417, tex2html_wrap_inline2419, tex2html_wrap_inline2421, tex2html_wrap_inline2423, tex2html_wrap_inline2425, tex2html_wrap_inline2427, tex2html_wrap_inline2429 and tex2html_wrap_inline2431.

Different techniques have been developed to calculate rotational velocities (e.g. Sletteback 1975; Tonry & Davis 1979; Gray 1992). Each one of these three methods relies to some extent on the geometrical technique, suggested originally by Shajn & Struve (1929), that relates line profiles and line widths to apparent rotational velocity, tex2html_wrap_inline2433. Collins & Truax (1995) noted that this technique implies the following three assumptions:

Under these assumptions, the standard rotation profile has the following form (Gray 1992)
where tex2html_wrap_inline2435 represents the maximum broadening and tex2html_wrap_inline2437 is the limb darkening coefficient. In this paper we will follow the method described by Gray (1992) since contrary to the Sletteback and Tonry & Davis methods which need to build up a calibration of rotational velocities according to some parameter (e.g. the FWHM in Sletteback), the Gray method provides direct and independent measurements of tex2html_wrap_inline2439. With this method the projected rotational velocity (tex2html_wrap_inline2441) is calculated from the Eq. (17.14) given in Gray (1992) which relates the frequencies where the Fourier transform of tex2html_wrap_inline2443 reaches a relative minimum and the wavelength of the line considered. This method is independent of the instrumental profile since the convolution of the intrinsic spectrum with the instrumental profile may add relative minimum values but will not change the position of the minima given by Eq. (17.14). Moreover, these minima introduced by the instrumental profile appear at much higher frequencies than those of the intrinsic spectrum used to calculate the rotational velocity. Projected rotational velocities of the observed stars are listed in Tables 3 and 4.

  Table 3: Rotational velocities of observed tex2html_wrap_inline2445 Sct stars. The fourth column indicates the value of v sin i obtained using the method described in Gray (1992). For La Palma spectra, the line chosen for the v sin i calculation appears in the last column. For McDonald spectra, the mean rotational velocity and the standard deviation are given in the forth column and the number of lines used is shown in the last column. The absence of suitable lines prevents the calculation of v sin i for VW Ari and CY Aqr

Table 3: continued

3.1. Estimated uncertainties in the calculated values of tex2html_wrap_inline2515

According to Eq. (1), one of the uncertainties in the calculation of tex2html_wrap_inline2517 comes from the limb-darkening law. In our calculations, we have assumed a linear limb-darkening law in the form
where tex2html_wrap_inline2519 with tex2html_wrap_inline2521 the angle formed by the line of sight and the direction of the emerging flux, I(1) represents the intensity at the stellar disk center and tex2html_wrap_inline2525, the linear limb-darkening coefficient, takes a value of tex2html_wrap_inline2527. Although some authors (Klinglesmith & Sobieski 1970; Manduca et al. 1977; Díaz-Cordovés & Giménez 1992) have proposed different non-linear laws for limb darkening, Díaz-Cordovés & Giménez (1992) also showed that the errors in the total emergent flux assuming a linear law are less than 2% in the range of temperatures where the tex2html_wrap_inline2529 Sct stars lie (tex2html_wrap_inline2531).

Wade & Rucinski (1985) have tabulated the linear limb-darkening coefficient in terms of wavelength, tex2html_wrap_inline2533 and tex2html_wrap_inline2535. Assuming an interval of temperatures tex2html_wrap_inline2537, tex2html_wrap_inline2539 and spectral ranges of tex2html_wrap_inline2541 (La Palma) and tex2html_wrap_inline2543 (McDonald) we obtain tex2html_wrap_inline2545 for La Palma spectra and tex2html_wrap_inline2547 for McDonald spectra.

To see to what extent the limb-darkening coefficient affects the calculated value of tex2html_wrap_inline2549, we have convolved a synthetic line (tex2html_wrap_inline2551), generated with ATLAS8 and with null rotational broadening by definition, with a grid of rotational profiles (tex2html_wrap_inline2553 = 30, 60, 90, 120, 150, 180 tex2html_wrap_inline2557) and we have calculated the rotational velocity for two values of tex2html_wrap_inline2559 (0.3, 0.6) which would correspond to the greatest difference in the McDonald spectra. The results are given in Table 5. We can see how the influence of the limb-darkening coefficient in the calculated value of tex2html_wrap_inline2561 increases when the rotational velocity increases. For McDonald spectra, where tex2html_wrap_inline2563 is always tex2html_wrap_inline2565 tex2html_wrap_inline2567, this effect is negligible. On the other hand, for La Palma spectra, where the difference (tex2html_wrap_inline2569) is, at worst, 0.07 this effect can also be neglected.

  Table 4: Rotational velocities for the sample of non-variable stars. The value of v sin i and the selected line(s) used for its determination are given as in Table 3


The determination of the local continuum is another unavoidable source of error: a displacement in the continuum level can change the line profile, especially the wings, and thus distort the shape of the Fourier transform and modifying the position of its zeroes. Despite of the excellent signal-to-noise ratio of the La Palma spectra, their moderate resolution and the high number of lines present in the spectral region considered make most of the lines appear blended which makes difficult the continuum placement. The error in the continuum determination for these spectra has been estimated by comparing equivalent widths of different lines in the observed spectrum of Procyon with those from the Atlas of Procyon (Griffin & Griffin 1979). An error in the continuum positioning of tex2html_wrap_inline2611 was adopted which correspond to an error of 8-10 tex2html_wrap_inline2615\ in tex2html_wrap_inline2617 for those stars with the highest rotational velocities, the error being lower when tex2html_wrap_inline2619 is lower. For McDonald spectra, where the continuum is much better defined, the error associated with the continuum level indetermination is negligible.

The equivalent width of the line also plays an important role in the accuracy of the tex2html_wrap_inline2621 value. This can be easily understood by considering that the peak of the main lobe in the transform corresponds to the equivalent width of the spectral line and the sidelobes are proportional to the main lobe. A large equivalent width will thus mean large amplitudes in the data transform and large relative height of the main lobe and sidelobes to the noise level.

The sampling frequency is another limiting factor in the calculation of tex2html_wrap_inline2623. Defining this frequency as tex2html_wrap_inline2625 and considering La Palma and McDonald spectral resolution we can get a lowest tex2html_wrap_inline2627 value of tex2html_wrap_inline2629 tex2html_wrap_inline2631 and tex2html_wrap_inline2633 tex2html_wrap_inline2635 for La Palma and McDonald spectra respectively. Hence, for stars with tex2html_wrap_inline2637 lower than these values it is not possible to calculate their rotational velocities but only an upper limit.

The intrinsic nature of tex2html_wrap_inline2639 Sct stars is another source of error: a pulsating star generates a velocity field in the line-forming region which manifests itself as a distortion in the profiles of spectral lines. Whereas radial pulsation will only produce a shift in the central wavelength of the spectral lines, velocity perturbations whose phase is a function of longitude (non-radial) will displace the contributions of the various longitudinal strips causing bumps and dips in the line profile (Walker et al. 1987). However and due to the small amplitudes of the tex2html_wrap_inline2641 Sct stars, the influence of the line distortions on tex2html_wrap_inline2643\ is negligible compared to other sources of error: Kennelly et al. (1992) measured the rotational velocities of a series of spectra of tex2html_wrap_inline2645 Boo obtaining an standard deviation of only tex2html_wrap_inline2647 tex2html_wrap_inline2649.

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