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

- The main source of broadening must be rotation thus Balmer lines and strong lines for which the pressure broadening is comparable with the rotational broadening must be excluded.
- Weak lines that could disappear with increasing rotational velocity must also not be used.
- The lines must lie in a spectral region which permits an accurate determination of the continuum level in order to avoid systematic errors in the measurements of the line width.
- The lines must be unblended.

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: and . The first one is stronger but it appears slightly blended with the weak line and it will be used only for where it is not possible to use the weaker line . 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: , , , , , , , and .

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, . Collins & Truax (1995) noted that this technique implies the following three assumptions:

- The observational aspect of a uniformly rotating star may be approximated by a circular disk subject to a linear limb-darkening law applicable to all parts of the stellar disk.
- The limb-darkening law for the line is the same as for the continuum.
- The form of the line does not change over the apparent disk.

where represents the maximum broadening and 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 . With this method the projected rotational velocity () is calculated from the Eq. (17.14) given in Gray (1992) which relates the frequencies where the Fourier transform of 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 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

According to Eq. (1), one of the uncertainties in the
calculation of comes from the limb-darkening law. In our calculations, we have
assumed a linear limb-darkening law in the form

where with 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 ,
the linear limb-darkening coefficient, takes a value of .
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 Sct stars lie ().

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

To see to what extent the limb-darkening coefficient affects the calculated value of , we have convolved a synthetic line (), generated with ATLAS8 and with null rotational broadening by definition, with a grid of rotational profiles ( = 30, 60, 90, 120, 150, 180 ) and we have calculated the rotational velocity for two values of (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 increases when the rotational velocity increases. For McDonald spectra, where is always , this effect is negligible. On the other hand, for La Palma spectra, where the difference () 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 was adopted which correspond to an error of 8-10 \ in for those stars with the highest rotational velocities, the error being lower when 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 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 . Defining this frequency as and considering La Palma and McDonald spectral resolution we can get a lowest value of and for La Palma and McDonald spectra respectively. Hence, for stars with lower than these values it is not possible to calculate their rotational velocities but only an upper limit.

The intrinsic nature of 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 Sct stars, the influence of the line distortions on \ is negligible compared to other sources of error: Kennelly et al. (1992) measured the rotational velocities of a series of spectra of Boo obtaining an standard deviation of only .

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