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6. Relationship between physical and pulsationparameters

6.1. Rotational velocity and amplitude

In Fig. 6.1 (click here) we have plotted the variation in amplitude of the observed tex2html_wrap_inline2889 Scutis against their rotational velocities. Although some authors (e.g. McNamara 1985) have suggested a value of tex2html_wrap_inline2891 as criterion to distinguish between low and large amplitude tex2html_wrap_inline2893 Sct stars, from this figure we can see how a value of tex2html_wrap_inline2895 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 tex2html_wrap_inline2901 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 tex2html_wrap_inline2903 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 tex2html_wrap_inline2905 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 tex2html_wrap_inline2909 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 tex2html_wrap_inline2911 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 tex2html_wrap_inline2913 tex2html_wrap_inline2915 (Dziembowski et al. 1988). Taking into account that the tex2html_wrap_inline2917 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 tex2html_wrap_inline2919 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 (tex2html_wrap_inline2921 =46 tex2html_wrap_inline2925, tex2html_wrap_inline2927) 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 tex2html_wrap_inline2929 is only of 4 tex2html_wrap_inline2931 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 tex2html_wrap_inline2933 Scuti but with a relatively small amplitude with respect to the rest of the stars within this category (tex2html_wrap_inline2935 =14 tex2html_wrap_inline2939, tex2html_wrap_inline2941), 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 tex2html_wrap_inline2943 Sct stars showing low amplitudes (tex2html_wrap_inline2945) and tex2html_wrap_inline2947 tex2html_wrap_inline2949 tex2html_wrap_inline2951 (GN And, V526 Cas, IM Tau and V644 Her) we could think that they may be large amplitude tex2html_wrap_inline2953 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 tex2html_wrap_inline2955 stars in the instability strip. However, the abundance analysis performed on this group of tex2html_wrap_inline2957 Scuti stars (Solano & Fernley 1997) does not indicate the presence of characteristics typical of tex2html_wrap_inline2959 stars (i.e. calcium underabundance and iron overabundance). More probably this group can be considered as normal low amplitude multimode pulsators tex2html_wrap_inline2961 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 tex2html_wrap_inline2963 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 tex2html_wrap_inline2967 Sct stars with an average value of tex2html_wrap_inline2969 tex2html_wrap_inline2971, the theoretical distribution gives tex2html_wrap_inline2973 stars with tex2html_wrap_inline2975 tex2html_wrap_inline2977 tex2html_wrap_inline2979 which is quite close to the observed number.



In order to find out whether the distribution of the tex2html_wrap_inline2987 values of the low amplitude tex2html_wrap_inline2989 Sct stars resembles that of non-variable stars, we proposed to apply the Kolmogorov-Smirnov test to our samples of low amplitude tex2html_wrap_inline2991 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 tex2html_wrap_inline2993).

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 tex2html_wrap_inline2995 Scuti stars are grouped around F0. Following Fukuda (1982) this would correspond to a difference of tex2html_wrap_inline2997 tex2html_wrap_inline2999\ in tex2html_wrap_inline3001. The fact that the non-variable stars show significantly lower values of tex2html_wrap_inline3003 than the low amplitude tex2html_wrap_inline3005 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 tex2html_wrap_inline3007 Sct stars with known tex2html_wrap_inline3009 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 tex2html_wrap_inline3011 Sct and the non-variable samples have the same number of stars (N=105) and show the same spectral type distribution. Those peculiar stars (tex2html_wrap_inline3015, tex2html_wrap_inline3017,...) which could have biased the tex2html_wrap_inline3019 distribution were not considered. Although two different catalogues have been used for the spectral type identification (Rodríguez et al. (1994) for the tex2html_wrap_inline3021 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 tex2html_wrap_inline3023 values of our observed stars (both tex2html_wrap_inline3025 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 tex2html_wrap_inline3027 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 tex2html_wrap_inline3029 Sct stars show a broader distribution in rotation velocity than the non-variable stars. Also, the average rotation velocity is higher for tex2html_wrap_inline3031 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 tex2html_wrap_inline3033 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 tex2html_wrap_inline3035 the cumulative distribution of tex2html_wrap_inline3037 Sct and non-variable stars are quite similar, the differences being present for tex2html_wrap_inline3039 tex2html_wrap_inline3041 tex2html_wrap_inline3043. This lead us to think that high rotation may increase the probability of tex2html_wrap_inline3045 Scuti pulsation.


6.2. Effective temperature and amplitude

In Fig. 6.1 (click here) we have plotted the effective temperature of both low and large amplitude tex2html_wrap_inline3071 Scuti stars versus the amplitude variation. In both cases, the tex2html_wrap_inline3073 values have been derived from our tex2html_wrap_inline3075 measurements. Moreover, for those large amplitude stars with tex2html_wrap_inline3077 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 tex2html_wrap_inline3079 photometry (mean tex2html_wrap_inline3081 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 tex2html_wrap_inline3083 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 tex2html_wrap_inline3085 Scutis compared to the low amplitude (Rodríguez et al. 1994) also support this hypothesis.

6.3. Period versus tex2html_wrap_inline3087 and effective temperature

In Fig. 6.3 (click here)a we show the distribution of periods for the tex2html_wrap_inline3089 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 tex2html_wrap_inline3091 Sct stars are evolved stars: from the period-mean density relation we get that tex2html_wrap_inline3093, 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 tex2html_wrap_inline3103 and effective temperature. Whereas no correlation is found between tex2html_wrap_inline3105 and period a clear relation is shown up between tex2html_wrap_inline3107 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 tex2html_wrap_inline3109 Sct stars.


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