## 5. Discussion## 5.1. The relevance of the diagramsEddington & Plakidis (1929) were the first to infer the
existence of cumulative errors in the periods of long-period
variables. Sterne (1934a,b) demonstrated that it is not necessarily
true that curvatures in the diagram
## 5.2. Is there a significant period change?In a first approach we applied Sterne's (1934b) procedure on the almost uninterrupted sequence of from to 280. This results in an estimate of the standard deviation of the measurement error and standard deviation of the intrinsic period scatter . The best all-purpose test for changes in the mean period we are aware of is the periodogram-based statistic of Lombard (1998). Fig. 6 shows the periodograms of the mean periods for the times of maximum and minimum. The lower panel (), especially, shows strong evidence for an increase in power at the lowest frequencies. Unfortunately, Lombard only presents forms of the test that are suitable for either complete datasets, or for incomplete sets with missing values which form a stationary sequence. Because the Cyg sequences of times of minimum or maximum are incomplete, a refinement of the method was necessary (see also the discussion in Koen & Lombard 1995). We adopted analogous estimators, and tested their adequacy by simulating 2000 datasets with realistic zero-mean Gaussian measurement errors ( d) and intrinsic period scatter ( d), using cycle numbers for the the times of maximum and minimum of Cyg. Our result establishes that the Lombard statistic is significant at better than the 5% level for the maxima, while the significance level for the minima is between 5% and 10%. We conclude that there is quite strong evidence for a changing mean period in the dataset comprising the times of maximum, while the support for a changing period is somewhat weaker in the other dataset.
## 5.3. What is the form of the period change?A variety of continuous time state space models as described by Koen (1996) were fitted to the two datasets. The Akaike and Bayes Information Criteria (see e.g. Koen & Lombard 1993) were used to compare the models. The best results, according to these criteria, were a deterministic, linear period change for the maxima, and no period change in the case of the minima. Both the optimal models also required non-zero intrinsic period scatter. The particularly simple solutions selected by the state space modelling technique imply that we can apply ordinary regression methods to the problem, provided that account is taken of the correlation between successive observations. We used the formulation in Koen (1996), Sect. 2, to find for the maxima ( being the period at the start of the observations). The model fit was assessed by checking the transformed residuals, as described by Koen (1996): the residuals were uncorrelated, mean-stationary and Gaussian. The best model for the minima had and the residuals again satisfied the relevant tests for randomness, stationarity and normality. Further models in which the intrinsic period scatter standard deviation was constrained to be the same as for the maxima (i.e. ), were also fitted to the Cyg minima. The best such model, again, had a constant period and the residuals' properties were satisfactory. A simple quadratic fit , where
over the complete range of available
data yields d y ## 5.4. The apparent cycles in the O-C diagramThe above suggests that there is a systematic linear period change
which is only detectable in the times of maximum, with its longer
baseline. Nonetheless, the pattern of cycles in the
diagrams is seductive, and it must be
checked whether there is any evidence for residual period changes once
the overall linear change has been prewhitened from the times of
maximum. We therefore studied the
## 5.5. The interpretation of the period changesMira stars are highly-evolved stars with strongly-reduced supplies
of nuclear fuel and dense carbon/oxygen cores surrounded by a growing
thin layer of helium. They represent a very short-lived phase in
stellar evolution, and in the H-R diagram they are located at the top
of the AGB. Contrary to popular belief, there is no evidence that
Miras systematically evolve to longer periods as they age (Whitelock
1996). When helium-burning begins, the helium shell expands till most
of the helium is consumed, the shell shrinks and the star returns to
hydrogen burning, and the building up of a new helium shell. This
process repeats a number of times, shell ignition and burning being
called the "shell flash". Continuous period variation are expected via
surface luminosity and radius changes: indeed, period changes
consistent with such a flash have been described by Wood & Zarro
(1981, herefater WZ), see also Gál & Szatmáry (1995)
for the Mira stars R Hya, R Aql, W Dra and T UMa.
WZ compared rates of period change with rates of changes predicted by
theoretical flash calculations. One of their stars, W Dra
(), shows an
diagram with similar characteristics
as seen in Fig. 4, though with a much stronger parabolic character and
with relatively less-pronounced short-term fluctuations (note that the
time base over which W Dra was observed was only about 100
cycles, or years). WZ place
W Dra between points B and C in their Fig. 1, which shows the
behaviour of the surface luminosity during a typical fully-developed
flash cycle. Barthès & Tuchman (1994) identified the
dominant pulsation mode of Cygni
with the first overtone, and derived a mass of
, which is in the range of the masses
used by WZ. We used this mass estimate in WZ's formulae with their
parameter values for first overtone pulsation, and calculated that the
luminosity variation corresponding to the period variation
amounts to
0.00002 y The interpretation outlined above, evidently, cannot account for any alleged short-term variations seen in the diagram. © European Southern Observatory (ESO) 1999 Online publication: December 22, 1998 |