Fig.4: Distribution of observed V/R periods.
Filled circles denote periods for stars
which show no evidence for binarity,
while open circles denote periods for stars
known as (or suspected to be) binaries.
Circles with upward arrows indicate that possible periods
are longer than the values shown.
Different periods in different epochs
for individual stars
are shown by data points connected by lines
For a total of 28 Be stars, Hirata & Hubert-Delplace (1981) showed that the period of the long-term V/R variation does not depend on the spectral type of the star. This conclusion was confirmed by Mennickent & Vogt (1991) for a total of 29 Be stars. Since the sample stars of Mennickent & Vogt (1991) only partially overlap those of Hirata & Hubert-Delplace (1981), we combined these two sets [we have missed four stars in the sample of Hirata & Hubert-Delplace (1981)], and also collected data for some other stars from the literature. Figure 4 shows the resultant distribution of the observed V/R (quasi-) periods for 53 Be stars of spectral type O9.5--B8. Filled circles denote periods for stars which show no evidence for binarity, while open circles denote periods for stars known as (or suspected to be) binaries (in a binary system, the companion star possibly affects the confinement of the m=1 oscilaltions through the tidal perturbation potential and the truncation of the disc). Circles with upward arrows indicate that possible periods are longer than the values shown in the figure. Some stars, which have exhibited different periods at different epochs, contribute two or more data points, which are connected by lines. The number of stars in each spectral bin is annotated at the top. For the sake of readers' convenience, we list detailed information for our sample stars in Table 1 in Appendix.
From Fig. 4, we observe the following features. First, as known before, the period of V/R variation shows no strong dependence on spectral type, although the sample of late-type Be stars is still scarce. Second, the data suggest a maximum around B3--B4. Third, the longer end of the period distribution is approximately independent of the spectral type, except for some stars with exceptionally long periods. Below we show that these features are consistent with the hybrid scenario of the confinement of the m=1 oscillations discussed in the previous sections. We should keep in mind, however, that about 1/3 of the Be stars have not shown long-term V/R variations in Balmer line profiles (Copeland & Heard 1963; Mennickent & Vogt 1991), which obviously could not be included in Fig. 4. The difference between these stars and V/R variables should be clarified by future studies.
First, we discuss the feature that
the observed V/R periods do not show strong spectral dependence.
As mentioned earlier,
the range of the rotation parameter f
of B5-type Be stars
is about twice as large as that of B0-type Be stars.
Since the value of 
for the former stars is likely
larger than that for the latter stars,
the deformation factor 
of B5-type Be stars would be
larger than that of B0-type Be stars by more than a factor of four.
Consequently,
if the confinement of the m=1 oscillations is attributed
solely to the rotational effect,
the periods of V/R variation of B5-type Be stars would be
shorter than those of B0-type Be stars by more than an order of magnitude
[compare Fig. 2(a) with Fig. 2(b)].
It is obvious that this does not fit the observed data.
Similarly, if the confinement is due
only to the radiative effect,
B0-type Be stars would exhibit V/R variations whose periods
are much shorter than those of B5-type Be stars.
Therefore, the observed absence of
a significant difference between V/R periods
for B0- and B5-type stars
is in agreement with the hybrid scenario of the confinement.
The rotational effect
is dominant in discs of B5-type Be stars,
while the radiative effect
plays an important role in discs of B0-type Be stars.
Based on this scenario,
the insensitiveness of the period to spectral type
is the result of the different mechanisms acting in
early- and late-type Be stars.
Fig.5: Schematic distribution of the V/R period
expected from the hybrid scenario of the confinement
of one-armed oscillations
Next, we show that the hybrid scenario of the confinement also
agrees well with the second and the third features
mentioned above.
As discussed in Sect. 4, the period of the m=1 oscillation
confined to the inner part of the disc
decreases with increasing values of
radiative parameter 
and deformation factor 
.
It is plausible to assume that the radiative parameter is, on average,
a monotonically increasing function of the effective temperature
of the central star.
Then, we expect that, among early-type Be stars,
the V/R period, on average, decreases towards earlier spectral type.
On the other hand,
as shown in Fig. 4 of Kogure & Hirata (1982),
the average value of rotation parameter f
monotonically increases towards later spectral type.
The deformation factor is, thus, larger for
a later spectral type,
unless the apsidal motion constant 
decreases with f
more rapidly than 
.
Hence, it is likely that, among late-type Be stars,
the V/R period decreases, on average, towards later spectral type.
As a result, as schematically shown in Fig. 5,
the distribution of the V/R period is expected to have a maximum
at a transitional spectral type around which the rotational effect
is comparable to the radiative effect.
Actually, the values of the radiative parameter,
the deformation factor, and
the parameter 
characterizing
the pressure effect would be different for individual stars.
Since the period of the m=1 eigenmode is sensitive to these values,
the distribution of the observed V/R periods is expected
to have a wide spread even for the same spectral type,
and to be limited below the critical period
above which no oscillations are confined.
The critical period would be roughly independent of
the spectral type, because
it depends mainly on the lower end of the distribution
of 
,
which is considered to be not so much different
between early- and late-type Be stars.
It should be noted that
these properties expected from the hybrid scenario of
the confinement of oscillations
agree well with the observed values
shown in Fig. 4.
It should also be noted that
in this scenario the V/R period can vary from one cycle to the next
for individual stars, which have been observed for some stars.
For late-type Be stars, the V/R period may change because of
a change in the magnitude of the pressure effect,
while for early-type Be stars it may change because of
a change in the combined radiative and pressure effects.
This prediction can be tested, and
will pose constraints on the model parameters.
Finally, we comment on the another observational aspect predicted by the hybrid scenario of the confinement. If the radiative effect is dominant, the V/R variations have periods independent of the value of the rotation parameter f. In contrast, if the rotational effect dominates, the period of the V/R variation should depend on the value of f. Thus, this hybrid scenario for the mechanisms which cause the confinement of the m=1 oscillations predicts that the periods of V/R variations of early-type Be stars have little correlation with the rotation parameter, while late-type Be stars should exhibit either the V/R variations with periods anticorrelated with the rotation parameter, or no V/R variation if the stars rotate too slowly.
Unfortunately, the data available at present are too scarce to
test this prediction over the whole spectral range.
It seems possible, however, to perform a rough test
for a restricted range of spectral types.
Table 1 lists 8 B1-type Be stars
of which the value of 
is known.
Three stars with shell spectra, 
Tau, V1294 Aql, and 59 Cyg,
have equatorial rotation velocities 
,
since stars with shell spectra are considered to be
seen nearly equator on.
The V/R periods for these stars range from 2 to 7 years.
On the other hand,
the V/R periods for
Aqr, o Pup, 8 Lac, and 25 Ori,
which have equatorial rotation velocities 
range from 2 to 10 years.
Therefore, no dependence of the V/R period on the rotation parameter
is seen for these B1-type stars.
Moreover, 48 Lib, which is the fastest rotator
among five B3-type stars with shell spectra
listed in Table 1,
does not show V/R periods shorter than
other four less-rapidly-rotating stars.
These results seem to support the hybrid scenario
of the confinement,
although we need more data for a definite conclusion.
