In Fig. 6.1 (click here) we have plotted the variation in amplitude of the observed Scutis against their rotational velocities. Although some authors (e.g. McNamara 1985) have suggested a value of as criterion to distinguish between low and large amplitude Sct stars, from this figure we can see how a value of represents better the border between these two subclasses.
Although from Fig. 6.1 (click here) we can see that both multimode (N=15) and single or double mode pulsators (N=10) are present in the low amplitude region, the presence of low amplitude single or double mode pulsators is possibly due to an insufficient number of observations: only with internationally coordinated campaigns that allow to have continuous data of a star over several weeks it has been possible to clearly identify the modes in small amplitude Sct stars (Ostermann et al. 1991; Kurtz 1994). Thanks to these campaigns, stars assumed to be single mode, radial pulsators in the past are now shown as non-radial pulsators (e.g. GX Peg, Michel et al. 1992). All the stars of Fig. 6.1 (click here) identified as low amplitude single or double mode pulsators were observed only from a single site and for intervals of time that range from a few hours (IK Peg, Wonnacot et al. 1994) to 10 nights (OX Aur, Gupta 1980). On the contrary, the stars identified as multimode pulsators typically correspond to continuous multi-site observations performed for longer intervals of time (e.g. GX Peg, Michel et al. 1992).
On the other hand, although it has been commonly accepted that, for large amplitude Scuti stars, only one or two frequencies appear excited and that both single and double modes reveal unequivocally radial pulsation (e.g., Rodríguez et al. 1992b; Garrido et al. 1990), some other authors have suggested the presence of non-radial modes in some large amplitude Scuti stars indicating that these stars might be multiperiodic with additional modes close to the limit of photometric detectability (e.g., Fu et al. 1995; Walraven et al. 1992).
The idea that Sct with large variations in amplitude show low rotational velocities, as it is shown in Fig. 6.1 (click here), was already pointed out by Danziger & Faber (1972) although their small sample did not let them draw any strong conclusion. Dziembowski (1980) suggested that, among low amplitude Scuti stars, the pulsational energy is shared among the different modes by the so-called non-linear mode coupling: the presence of radial and non-radial modes permits the effective sharing of energy through resonances which prevent a particular pulsation mode from developing a large amplitude. This author also showed theoretically that the three-mode coupling significantly reduces the amplitude for stars with rotation velocities (Dziembowski et al. 1988). Taking into account that the values derived from La Palma spectra for large amplitude stars cannot be regarded as real values but only upper limits, (Sect. 3), and relying only on those values calculated from the McDonald sample, we can see that the limit in rotation velocity predicted by Dziembowski corresponds well with the observed limit. V1162 Ori ( =46 , ) may be an exception. However, the monoperiodicity of this star needs to be more firmly established. Although due to the low signal-to-noise ratio only four lines were used to derive the rotation velocity, the value of the standard deviation in is only of 4 which indicates that there are not important differences between the values derived from each line and the calculated rotation velocity being, thus, real. Moreover, the fact that CC And, catalogued as large amplitude Scuti but with a relatively small amplitude with respect to the rest of the stars within this category ( =14 , ), also exhibits radial and non-radial modes could make us consider an intermediate stage where non-radial modes and large amplitudes can coexist.
Considering the four Sct stars showing low amplitudes () and (GN And, V526 Cas, IM Tau and V644 Her) we could think that they may be large amplitude Sct stars whose amplitudes have been limited by some physical mechanism. An efficient way of reducing the pulsational amplitude is to reduce the amount of helium in the helium ionization zone: according with the diffusion theory (Baglin 1972), in a stable atmosphere helium tends to sink due to gravity which is not balanced by radiation pressure. This is the commonly adopted mechanism to explain the presence of metallic stars in the instability strip. However, the abundance analysis performed on this group of Scuti stars (Solano & Fernley 1997) does not indicate the presence of characteristics typical of stars (i.e. calcium underabundance and iron overabundance). More probably this group can be considered as normal low amplitude multimode pulsators Scuti stars seen pole-on. IM Tau is, in fact, a multimode pulsator and GN And is assumed as single pulsator but probably because of an insufficient number of observations (only three nights, Rodríguez et al. 1993b). No frequency analysis is available for V526 Cas and V644 Her. This suggestion is reinforced if a theoretical distribution assuming random orientation of the rotation axes and Maxwellian distribution is considered (Gray 1988). In our case, for a sample of N=51 low amplitude Sct stars with an average value of , the theoretical distribution gives stars with which is quite close to the observed number.
In order to find out whether the distribution of the values of the low amplitude Sct stars resembles that of non-variable stars, we proposed to apply the Kolmogorov-Smirnov test to our samples of low amplitude Scutis and non-variable stars. We decided to use this test (K-S test, hereafter) since it treats the individual observations separately, thus ensuring that no information is lost because of binning (unlike other tests like the ).
It is well known that the A-F spectral types constitute a transition region in the distribution of rotational velocities of dwarf stars: According to Fukuda (1982) the rotational velocity is large for hot stars, it keeps constant for normal A stars dropping rapidly through the F-star region and is small for cool stars. This will cause a bias effect if both samples of stars do not show similar distributions of spectral types. In Fig. 6.1 (click here) we can see how the observed non-variable stars reveal a peak around F5 spectral types whereas the low amplitude Scuti stars are grouped around F0. Following Fukuda (1982) this would correspond to a difference of \ in . The fact that the non-variable stars show significantly lower values of than the low amplitude Sct stars will affect the cumulative distributions functions making the K-S test not valid. To solve this problem, we have compared only the normal A-stars in both samples (Fig. 6.1 (click here)). However, there are only 6 normal A-stars in the non-variable sample and for the sake of statistical significance, we would like to have data samples as large as possible. In an attempt to improve this situation, we have added those low amplitude Sct stars with known given in Rodríguez et al. (1994). At the same time, the sample of non-variable stars has been completed with stars from Hoffleit (1982) in such a way that both the low amplitude Sct and the non-variable samples have the same number of stars (N=105) and show the same spectral type distribution. Those peculiar stars (, ,...) which could have biased the distribution were not considered. Although two different catalogues have been used for the spectral type identification (Rodríguez et al. (1994) for the Sct stars and Hoffleit (1982) for the non-variable stars) we did not find significant differences between them after comparing the spectral types of the set of stars in common with both catalogues. On the other hand, the values of our observed stars (both Sct and non-variables) were calculated following the method described in Sect. 3 whereas the values of the stars added to both samples were taken from Uesugi & Fukuda (1982). A comparison between our values and those given in Uesugi & Fukuda (1982) does not yield systematic differences which could affect the statistical analysis. In Fig. 6.1 (click here) we compare the histograms of both samples as well as their correspondent cumulative distribution functions. The null hypothesis adopted in the K-S test (the two data set are drawn from the same distribution function) can be rejected with a 98% of confidence level. In this figure we can see that the Sct stars show a broader distribution in rotation velocity than the non-variable stars. Also, the average rotation velocity is higher for Sct stars than for non-variable stars which agrees with Breger (1979). Although this author suggested that the differences may be caused by the inclusion of stars this is not our case since only normal A-star have been considered. Moreover, in Fig. 6.1 (click here) we can see that up to 100 the cumulative distribution of Sct and non-variable stars are quite similar, the differences being present for . This lead us to think that high rotation may increase the probability of Scuti pulsation.
In Fig. 6.1 (click here) we have plotted the effective temperature of both low and large amplitude Scuti stars versus the amplitude variation. In both cases, the values have been derived from our measurements. Moreover, for those large amplitude stars with photometry available in the literature, the whole range of temperatures over a pulsation cycle has been also calculated using the MD85 calibration. We have applied the K-S test to compare the effective temperatures of the samples of low amplitude and large amplitude with photometry (mean along the cycle). Although due to the small number of large amplitude stars considered we cannot conclude from the K-S test that both samples are drawn from different distributions, Fig. 6.1 (click here) seems to indicate that large amplitude stars tend to have lower effective temperatures. This result is consistent with Breger (1980) who suggest that large amplitude Sct stars are more evolved stars crossing the Instability Strip at higher luminosities and therefore, on average, cooler temperatures. The scarcity of the large amplitude Scutis compared to the low amplitude (Rodríguez et al. 1994) also support this hypothesis.
In Fig. 6.3 (click here)a we show the distribution of periods for the Sct stars given in Rodríguez et al. (1994). In the multiperiodic stars we have only considered the period which corresponds to the dominant radial pulsation mode in every case. In this figure we can see that large amplitude stars tend to have longer periods: the K-S test predicts that the hypothesis that both samples are drawn from the same distribution can be rejected with a 99.9% of confidence level. Although, as Rodríguez et al. (1994) pointed out, there may be in this figure selection effects since low amplitudes are difficult to detect in stars with long periods because of the photometric stability required to get precise light curves, these results agrees with the idea proposed by Breger that large amplitude Sct stars are evolved stars: from the period-mean density relation we get that , where P is the period and L, M and T the luminosity, mass and effective temperature of the star respectively. From this relation we can see that the decrement in temperature of the post- main sequence phase of evolution would produce an increment of the period.
In Fig. 6.3 (click here)b we also plotted the relations between the period and and effective temperature. Whereas no correlation is found between and period a clear relation is shown up between and period: the period is shorter when the temperature is higher. This relation can be easily understood in terms of the amplitude-temperature- period relation described above: low amplitude tend to have higher temperature and shorter periods than large amplitude Sct stars.