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Astron. Astrophys. 342, 167-172 (1999)

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5. Discussion

5.1. The relevance of the [FORMULA] diagrams

Eddington & 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 [FORMULA] diagram always indicate period variations: such deviations can also be caused by the cumulative effect of random errors in the lengths of the cycles of the star itself. In the case of Mira stars, for example, Koen (1992) even showed that white noise could lead to an [FORMULA] plot with apparently significant indications of the presence of an entirely spurious period variation. The question thus arises whether the interesting structures revealed in Figs. 3 and 4 do stand for changes of the pulsation period.

5.2. Is there a significant period change?

In a first approach we applied Sterne's (1934b) procedure on the almost uninterrupted sequence of [FORMULA] from [FORMULA] to 280. This results in an estimate of the standard deviation of the measurement error [FORMULA] and standard deviation of the intrinsic period scatter [FORMULA].

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 ([FORMULA]), 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 [FORMULA] 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 ([FORMULA] d) and intrinsic period scatter ([FORMULA] d), using cycle numbers for the the times of maximum and minimum of [FORMULA] 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.

[FIGURE] Fig. 6. Periodograms of the mean periods for the times of maximum and minimum.

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 ([FORMULA] 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. [FORMULA]), were also fitted to the [FORMULA] Cyg minima. The best such model, again, had a constant period


and the residuals' properties were satisfactory.

A simple quadratic fit [FORMULA], where [FORMULA] over the complete range of available data yields [FORMULA] d y-1, implying that [FORMULA] Cygni's pulsation period has increased by about 4 days or 1% since the discovery date three centuries ago.

5.4. The apparent cycles in the O-C diagram

The 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 [FORMULA] 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 untransformed , (i.e. raw) residuals after subtraction of the linear trend. A little thought shows that Lombard's theory as described above applies also to these residuals. The periodogram of the residuals is shown in Fig. 7: there still appears to be an excess of low-frequency power. However, the residuals' statistics attain significance levels of around 10%, which is not entirely convincing: it may be, for example, that the global period change is slightly nonlinear, so that prewhitening by a linear trend is not quite appropriate.

[FIGURE] Fig. 7. Periodogram of the residuals after subtraction of a linear period trend (times of maximum).

5.5. The interpretation of the period changes

Mira 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 ([FORMULA]), shows an [FORMULA] 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 [FORMULA] 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 [FORMULA] Cygni with the first overtone, and derived a mass of [FORMULA], 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 [FORMULA] amounts to 0.00002 y-1, about 20 times less than WZ's result for W Dra. The three-centuries time span of the [FORMULA] Cyg data, and the low gradient of the luminosity change is compatible with a location of this star after point E in WZ's Fig. 1-that is, during the rise in luminosity when the H shell takes over as the main energy producer. We then calculated [FORMULA] as a function of time for a sequence of core masses [FORMULA] to 0.9, and find that this rate of luminosity change occurs at about 0.15-0.20 of the total flash cycle length after point E. We stress that the calculated luminosity-change rate and the derived epoch within the flash cycle are not critically sensitive to numerical values of the parameters entering the equations. Indeed, using the extreme values of the various coefficients in WZ and the lowest quoted mass ([FORMULA], Barthès 1998) will result in only a small variation of [FORMULA].

The interpretation outlined above, evidently, cannot account for any alleged short-term variations seen in the [FORMULA] diagram.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998