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.
and effective temperature
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.
