3 Results on the individual variables

3.1 Changing the type of variability: Y Persei

Y Per is a well-known carbon Mira star, though its period (about 250 days) is the shortest one among them (Groenewegen et al. 1998). There was no indication of peculiarity until 1987, when its amplitude dropped significantly. Furthermore, it fits exactly the PL relation of galactic carbon LPVs (Bergeat et al. 1998), thus this star has been considered as a typical member of its type.

However, as has been pointed out in Paper I, the appearance of the visual light curve changed dramatically around JD 2447000 (1987). The earlier Mira-type variations disappeared and were replaced by a semiregular and low-amplitude brightness change. In order to trace the time-dependent variations and quantify the sudden change, we performed a detailed lightcurve analysis utilizing subsets of the whole dataset.

The finally merged and averaged light curve covers 28 500 days (more than 110 cycles). There are smaller gaps in the fist half of the data, while the second half is completely continuous. We divided the data into eight 3000 day-long segments (each containing about 12 cycles) and one 4500 day-long segment. This enabled an accurate period determination in every segment avoiding the possible period smearing due to its long-term variation. The period was calculated with the conventional Fourier-analysis and was checked independently with a non-linear regression analysis.

Data in the first eight segments can be described very well with only one harmonic, but the last one with a two-component harmonic sum. None of the residuals shows significant periodic signals. The fitted curves are plotted in Fig. 1, while the resulting parameters are presented in Table 2. Although the formal standard errors are quite small, the real uncertainties are a little larger, most probably due to the intrinsic variations of Y Per. Therefore, we adopted a period uncertainty of $\pm1$ day and an amplitude error of $\pm0.05$ mag.

   

Table 2: Frequencies, the corresponding periods and amplitudes in nine subsets

No.	$f (10^{-3}{\rm c/d})$	$P ({\rm day})$	$A ({\rm mag})$
	(s.e.)		(s.e.)
1	3.950 (0.080)	253.2	0.67 (0.09)
2	3.915 (0.009)	255.4	0.81 (0.03)
3	3.954 (0.007)	252.9	0.69 (0.03)
4	3.930 (0.012)	254.5	0.63 (0.04)
5	3.953 (0.006)	253.0	0.91 (0.03)
6	3.959 (0.004)	252.6	1.07 (0.03)
7	4.023 (0.005)	248.6	0.89 (0.02)
8	4.058 (0.005)	246.4	0.80 (0.02)
9	4.095 (0.005)	244.2	0.36 (0.02)
	7.864 (0.012)	127.2	0.16 (0.02)

The amplitude and period changes seem to be well correlated as shown in Fig. 2, which is a well-known property of non-linear oscillators. A further interesting point is that the secondary period occuring in the last segment is exactly the half of the earlier dominant period within the error bars (the mean period in the first six segments is 253.8 days). Although it can be pure numeric coincidence, we will shortly discuss the possible relevance of this period halving.

Figure 2: The variations of the dominant period and amplitude in the 3000 day-long segments

As has been mentioned in Paper I, the abruptness of this amplitude change is quite surprising. The characteristic time scale of the amplitude decrease in other stars (V Boo - Szatmáry et al. 1996, R Dor - Bedding et al. 1998, RU Cyg - Paper I) varies from few hundred to few thousand days, typically tens of pulsation cycles. In Y Per the change happened in 400 days, between JD 2447200 and 2447600, or less than 2 cycles. Adopting a pulsational approach, it can be interpreted as a fast appearance of a new mode beside a slightly changing dominant mode. Recent models by Xiong et al. (1998) suggest a first and third overtone combination (observed periods: 245 and 127 days; theoretical prediction: P₁=231 days, P₃=130 days). According to Xiong et al. (1998), the coupling between convection and pulsation depends critically on the ratio of the timescale of convective motion and that of pulsation. The effect is stronger for overtone oscillation, as the turbulent viscosity becomes the main damping mechanism of the high overtones (it converts the kinetic energy of ordered pulsation into random kinetic energy). There have been a considerable number of theoretical and observational efforts in order to quantify this coupling and to detect observables related to it. Anand & Michalitsanos (1976) have already formulated a simple nonlinear model assuming that the convective envelope of M giants is composed of giant convection cells comparable in size to the stellar radius. They showed that the coupling can produce asymmetric fluctuations of the entire star. Such large perturbations may cause also fast changes in the pulsational properties. The earlier theoretical models are supported by modern high-resolution spectroscopic observations. Nadeau & Maillard (1988) observed velocity gradients of a few km s^-1 for lines of different excitation potentials, which was interpreted as being caused by convective motions. Most recently, Lebzelter (1999) presented such radial velocity measurements of semiregular variables, which implied the possibility of large convective cells with radial motions close to a few km s^-1. A simple estimate of the convective time scale can be found from the ratio of the thickness of the convective envelope and the observed order of magnitude of convective velocity. For 100 $R_\odot$ and 1 km s^-1 one get a result of 800 days being close to the duration of sudden changes in SRVs. Unfortunately, present instability surveys cannot calculate the amplitude of pulsation and, consequently, its long-term variation can be interpreted only speculatively.

We note that the first and third overtone model is very similar to that derived for R Dor by Bedding et al. (1998). They concluded in this instance that the rapid changes can be described very well with chaotic effects discussed by Icke et al. (1992). Furthermore, the period halvings noted above might be another hint for the presence of weak chaos. It can be interpreted as an inverse process of the period doubling bifurcation which may happen just beyond the onset of chaos (Kovács & Buchler 1988). It has been shown in a number of papers (e.g. Kovács & Buchler 1988; Saitou & Takeuti 1989; Moskalik & Buchler 1990), that different stellar pulsation models (W Vir, RV Tau, yellow semiregulars) show period doubling bifurcation leading from a regular to a quite irregular variation. Although the behaviour of Y Per does not fit exactly the proposed way of transition from regular to irregular state through cascades of period doubling bifurcations (and simultaneous noisy period halvings), this kind of explanation cannot be excluded.

Finally, the conventional classification of long period variables (LPVs) into Mira and SRa, SRb, and SRc type semiregular variables, as given in the GCVS, is not based on physical parameters and does not adequately cover even the behavior of some of the brightest and most studied LPVs. This has been improved by Kerschbaum & Hron (1992) by involving the "blue'' and "red'' subgroups of semiregulars. The SRa stars seem to form a mixture of intrinsic Mira and SRb variables. For Mira stars no similar definition using various stellar parameters exists. Unfortunately, the fact that Y Per does not fit neither of main types of LPVs (Mira and semiregular), does not uncover the underlying physical processes being responsible for its peculiarity.

3.2 Amplitude modulation I.: Beating in RX Ursae Majoris andRY Leonis

In Paper I we presented eight stars with closely separated frequencies. A few of them are triply periodic stars, in which mode changes occur quite frequently (see Sect. 3.4) and the close frequency components do not exist simultaneously. However, we found two variables with stable biperiodic variation, where various subsets show the same frequency content (V CVn and RY Leo). In those cases the amplitudes of components are quite different, therefore, no clear beating occurs, as in RX UMa. We plotted a typical subset of the light curve of RX UMa in Fig. 3 where a three-component fit is also shown ( f₀=0.005000 c/d, A₀=0.37 mag, f₁=0.005288 c/d, A₁=0.26 mag, f₂=0.010236 c/d, A₂=0.16 mag). The frequency spectrum of the whole dataset is plotted in Fig. 4.

If these frequencies have pulsational origin, then their ratios ( f₁/f₀=1.06, f₂/f₁=1.93) may give information about the mode of pulsation. Period ratios near 1.9 are very common in semiregular variables (see Paper I and Mattei et al. 1998) which was identified in case of R Dor to correspond to the first and third overtones (Bedding et al. 1998). Period ratios close to 1 may suggest high (3-5th) overtones, but theoretical calculations suggest strong damping in this mode domain (Xiong et al. 1998). Mantegazza (1988) found similar close frequency doublet in the red semiregular Z Sge and using models by Fox & Wood (1982) speculated about the possibility of second and third overtones. It is also possible, that one of the frequencies correspond to a non-radial mode (e.g. Loeser et al. 1986), but not much is known about the non-radial pulsation of red giant stars. In the other two stars mentioned earlier there are also two closely-spaced periods, which are demonstrated by the DFT spectra in Figs. 5-6. Note, that in V CVn even the cross-production terms ( $f_0\pm f_1$) are present unambiguously.

Figure 3: The observed and fitted light curves of RX UMa

Figure 4: The frequency spectrum (top) and window spectrum (bottom)

3.3 Amplitude modulation II.: Pulsation + rotation in RY Ursae Majoris?

Figure 5: The frequency spectrum of V CVn

Figure 6: The frequency spectrum of RY Leo

The amplitude modulation discussed in the previous subsection can be simply described by the beating of two close frequencies, since the mean brightness did not change with time in those stars. However, RY UMa (and partly RS Aur) shows amplitude variations which highly resembles those observed in RR Lyrae variables with Blazhko effect, where the minimum brightness changes much more significantly, than the maximum brightness. Therefore, a significant mean brightness variation can be observed.

Figure 7: The observed light curve (10-day means) of RY UMa

This interesting light curve variation has been highlighted in Paper I, where a 9800-day segment was analysed. In the meantime, historical AAVSO observations were added, extending the light curve to a whole length of 17000 days. Unfortunately, the early light curve during the first 3000 days has a less dense coverage, thus the most homogeneous data cover about 14000 days. The corresponding averaged light curve is shown in Fig. 7. In our sample of 93 stars studied in Paper I this behaviour is quite rare, only RS Aur seems to have similar light curve phenomenon (Fig. 8).

We have tried to explain the observed amplitude modulation with rotational effects. Earlier theoretical studies have generally neglected the stellar rotation, since typical rotational periods of red giants, usually obtained theoretically, are about 4000-10000 days, much longer than the characteristic times of pulsation. However, RY UMa shows such complex light variation and frequency spectrum (see later), that a possible explanation could be the rotation-pulsation connection. The amplitude variations turned out to be highly repetitive, which is shown in Fig. 9, where the light curve has been folded with two periods ( $P_{\rm pul}=306$ days, $P_{\rm mod}=4900$ days  $\approx 16\cdot P_{\rm pul}$).

We have tried to build a simple model, which involves a rotationally modulated non-radial oscillation. Our very approximate model consists of: i) a distorted stellar shape caused by a low-order non-radial oscillation; ii) stellar rotation with a period of 9800 days (i.e. twice of the period of modulation); iii) simple limb darkening (u=0.6). We have considered a triaxial ellipsoid having a short axis of unity, while the other axes change sinusoidally in time between 1.0-2.0 and 1.75-2.0 with the period of pulsation (306 days). The whole ovoid rotates and the intensity is integrated over the surface elements assuming normal limb darkening with coefficient u=0.6. The light curve is calculated as the logarithm of the change of the surface facing the observer.

Figure 8: The observed light curve (10-day means) of RS Aur

Figure 9: The light curve of RY UMa folded with the periods of pulsation (top) and modulation (bottom)

All aspects of this approach can be, of course, easily challenged. The introduced distorsion in our model is much larger than that of predicted by traditional description of non-radial oscillations. However, there have been a number of high-resolution observations (imaging and interferometry) of nearby Mira variables showing substantial asymmetric structures (Tuthill et al. 1999; Lopez et al. 1997; Karovska et al. 1997), which imply the incompleteness of spherical assumptions. Furthermore, fully three dimensional and turbulent dynamic numerical simulations of red giant stars (Jacobs et al. 1998) also suggest occurence of bipolar atmospheric motions and distorted stellar (photospheric) shape. Thus, we conclude, that it may be possible that the assumed maximum oblateness for RY UMa is not completely unlikely.

The second aspect is stellar rotation. Asida & Tuchmann (1995) explored theoretically the asymmetric mass-loss from rotating red giant variables and presented a scenario for an anisotropic mass ejection from AGB variables caused by rotational effects. Further support of rotationally induced variations in red giants was given by Barnbaum et al. (1995), who suggested a possible connection between rapid rotation ( $P_{\rm rot} \approx$ 530 days) and pulsation in the carbon star V Hya (type SRa). In our case there is no need for assuming unusual rotation, since the required rotational period (9800 days) is in the range of what we expect for such extended and evolved objects as red giants.

The weak point in our model is the neglect of temperature variations along the pulsation and the assumption of constant (and solar) value for the limb-darkening coefficient. The latter is less significant, because even completely neglecting the limb darkening (i.e. using a uniform disk) does not change the calculated light curve significantly. This is especially fortunate keeping in mind dynamical model analyses by Beach et al. (1988), performed in order to determine limb darkening/brightening function for Mira atmospheres, which illustrated that the usual limb darkening correction of uniform disk model in lunar occultation measurements can be even in the wrong direction in certain pulsational phases. The temperature variations could not be taken into account, since we have no information neither about its range nor about its phase dependence. The strong non-radial assumption would imply weaker temperature effects, but without any kind of phase dependent temperature measurements, we cannot draw a firm conclusion. Nevertheless, our main purpose is rather to get a qualitative "fit'' of the observed behaviour than to quantitatively model a visual light curve. A graphical representation of the model and the resulting light curve is given in Fig. 10.

Figure 10: The calculated model light curve and selected phases with different geometric aspects

The resulting model light curve was compared with the observed one through their frequency spectra. This comparison can be seen in Fig. 11, where the corresponding Fourier spectra are shown.

Figure 11: Discrete Fourier-transforms of the modelled (top) and observed (middle) light curves. Bottom panels show the window functions. There are apparent splittings around the corresponding peaks in the frequency spectra, which are (model) and might be (observations) due to the stellar rotation

Beside the overall similarity the most striking feature is the frequency splitting of the main component at f₀=0.003268 c/d. Such a splitting has been well-known behaviour in pulsating white dwarfs and roAp stars (e.g. Shibahashi & Saio 1985; Buchler et al. 1995; Kurtz et al. 1996; Baldry et al. 1998), while recently it has been detected in several RRab stars showing Blazhko effect (e.g. Kovács 1995; Nagy 1998; Chadid et al. 1999) and in a number of MACHO RRc variables (Alcock et al., in preparation). One of the commonly accepted views is the oblique pulsator model, where a non-radial oscillation is coupled with the rotation, as the rotational axis does not coincide with the symmetry axis of pulsation (Shibahashi 1999). In the case of RY UMa, a frequency triplet is present with $\Delta f \approx 0.0002$ c/d, which correspond to the frequency of amplitude modulation. Also, the triplet has an asymmetric amplitude distribution with respect to the central frequency peak. Similar asymmetry can be attributed to intrinsic nonlinear mechanisms (Buchler et al. 1995), that seems to be quite likely in a highly non-spherical environment. We note the presence of a peak at 0.00273 c/d corresponding exactly to one year. This is little surprising, because the light curve has no seasonal gaps (it is circumpolar from Europe, Japan and North America), therefore a such alias peak is not expected. A possible reason for this could be the seasonal increase of the scatter, since despite its circumpolarity it is more difficult to observe near the horizon.

We give further illustration of the amplitude variations by the wavelet transform shown in Fig. 12.

Figure 12: The wavelet map of RY UMa. The main ridge of the primary frequency component (indicated by the small arrow) was sliced to get the time-dependent amplitude values plotted in Fig. 13

This time-frequency analysing method is quite successful in quantifying the time-dependent variations in astronomical time series (see, e.g. Szatmáry et al. 1994; Szatmáry et al. 1996; Foster 1996 and Paper I). The overall pattern is quite systematic suggesting the regularity of the underlying process(es). The amplitude of the main ridge has been extracted along the time axis in order to plot its variations in Fig. 13.

Figure 13: The normalized amplitude variation of RY UMa from the wavelet map

Our conclusion is that despite the local irregularities of the visual light curve (see the larger scatter in the bottom panel of Fig. 9 between $\phi=0.6-0.8$), the amplitude modulation implies a regular physical process, e.g. rotation or binarity. In this paper we adopted to rotation, however, other possibilities cannot be excluded using the presently available observations.

3.4 Amplitude variations due to repetitive mode changes: W Cygni, AF Cygni

W Cyg and AF Cyg are two of the most popular and well-observed semiregular stars (e.g. Percy et al. 1993, 1996). These stars are two illustrative examples of repetitive mode changes, where some modes turn on and off on a time scale of few hundreds of days (a few cycles). This has been partly highlighted in Paper I, where TX Dra and V UMi were discussed in terms of varying dominant modes. In this paper we focus on two other stars, which further supports our belief that this phenomenon may be quite frequent in semiregular variables.

Both stars have continuous light curves covering more than 28000 days, which corresponds to 300 and 215 cycles for AF Cyg and W Cyg, respectively, considering their shorter periods (93 and 130 days). There are several occasions when their light curves completely change. This is illustrated in Figs. 14-15, where we plotted two 1000-days long subsets for both stars.

Figure 14: 1000-days long noise-filtered (Gaussian smoothing with an FWHM of 8 days) subsets of W Cyg showing different states with different dominant modes. Note, that solid line only connects the points and does not represent any kind of fits

Figure 15: The same as in Fig. 14 for AF Cyg

The photometric behaviour (cycle lengths and their amplitudes) changes dramatically from time to time. The wavelet map of W Cyg (Fig. 16) and the cross sections of the main ridges (Fig. 17) clearly illustrate the amplitude variations.

Figure 16: The wavelet map of W Cyg

We obtained similar results with AF Cyg.

There are no obvious indications for periodicity in Fig. 17. However, a few important points can be drawn. The first is that the plotted amplitude changes should be considered real, as the light curve has no gaps after JD 2421000 (1916). Therefore, the repetitive amplitude decreases and increases are not numerical artifacts caused by the inappropriate data distribution (see Szatmáry et al. 1994 for testing the method with simulated light curves). The amplitude of the dominant mode ("mode 1'', P=130 days) changes on a time scale of 2000-3000 days. This may imply that the exciting mechanism of pulsation is intermittent and the damping is strong. The bimodal state (i.e. simultaneous high amplitudes of "mode 1'' and "mode 2'') is quite rare. Furthermore, only weak hints are present for simultaneous or alternating modes suggesting different and independent excitations for the two modes corresponding to the first and third overtones in theoretical models of Xiong et al. (1998) - P₁=231 days, P₃=130 days.

As has been mentioned above, AF Cyg has similar unstable behaviour, only the periods (and possibly the modes) are different. The two main periods (163 days and 93 days) would suggest second and fourth overtones adopting models of Xiong et al. (1998) (P₂=173 or 154 days, P₄=103 or 91 days).

Figure 17: The normalized amplitude variations of the two main modes

Finally, episodic amplitude and period changes were also reported for three semiregulars (RV And, S Aql, U Boo) by Cadmus et al. (1991), where the dominant period was the shorter one during the low-amplitude epsides. In addition, Mattei & Foster (2000), who studied long-term trends of period, amplitude, mean magnitude and asymmetry in the AAVSO light curves, reported several stars with such trends. But what we see here is completely different in W Cyg and AF Cyg, where the shorter period is mainly the dominant one with higher amplitude.