The smooth, simple structure of the period change curves for both modes seems to rule out the presence and importance of any kind of stochastic processes suggested by Balázs-Detre & Detre (1966) as an explanation of complex structure of curves. There are no large period change rates that would need the mixing events associated with the semiconvective zone of the stellar core as an explanation (Sweigart & Renzini, 1979). However, the abrupt period increase of the first overtone of v54 needs the mixing events theory. According to Lee (1991) the observations of period changes are consistent with his evolutionary models only if the observational errors are of order days/Myr. For his model , while after the addition of a random error the dominant range of period change is for M 15. Now we can determine the period changes with higher precision, and then the distribution of period changes disagrees with the evolutionary prediction.
However, Lee's evolutionary model is derived independently from the type of pulsation mode for single mode RR Lyrae stars. At the present view the sign and less definitely the amount of the period change rate, independently from the type of pulsation ( - fundamental, - first overtone), are closely connected to the direction of evolution on the HRD. The period change behaviour of the different modes of multimode pulsators have been theoretically investigated for AC And models (Kovács & Buchler, 1994). Both theoretical and observational period changes were found to be the same in size and sign for AC And. Different sizes of period changes for two modes in 4 CVn (Breger, 1990) were reported. The sign of period changes were the same for both modes, although it contradicted to the theoretically predicted direction.
The most striking result of the present investigation, the very definite period change rates of different sign for the fundamental and first overtone modes in the double mode RR Lyrae stars, v53 and v17, in M 15 suggests that caution is needed in connecting the sign of period change rate and the direction of the star's evolution in the HR diagram.
Whether the period changes of the different types of pulsation modes have special regularities during the evolution and the different modes of the double mode pulsators have the same regularities as the different types of single mode pulsators or not is open for theoretical investigations.
However, on the observational basis we can compare our knowledge of the different period change behaviour of and stars, which was neglected in the last years, with that of the fundamental and first overtone modes of stars in M 15.
The difference in the period change behaviour that is associated with the pulsation mode was first reported by Szeidl (1975). Barlai (1977) did note that type variables (fundamental mode) tended to show larger period changes (both positive and negative) than did the type stars (first overtone). Smith & Sandage (1981) also found that the type variables exhibit a greater range of period changes than do the types but, as he remarks, the difference in the mean rate of period change for and types variables is too small to be significant. However, Silbermann and Smith (1995) reported that the standard deviation about the mean for the and values are more than twice as large for the variables as for the stars. The wider range for the variables is also evident in their histogram of the rate of period change (see their Fig. 3.).
Comparing the result of the present investigation for the period change rate of the fundamental mode in stars to stars, we emphasize the significantly larger period change rate for the fundamental mode than for the first overtone mode (Table 4. and Fig. 4.). There is, however, a definite difference in the sign of the period change rates. According to Silbermann and Smith (1995) the mean value of for stars equals +0.08 0.10 while the fundamental modes of in M 15, with except one star, have negative sign of period change rates.
According to Silbermann and Smith (1995) the rates of period changes observed for the eight variables are similar to those of variables (: =+0.04 0.02 and : =+0.05 0.02). If we take into account that stars are pulsating in first overtone and that in stars the dominant mode is the first overtone, the agreement would suggest some kind of regularity for the first overtone in both and stars.
Although our results of for the first overtone of stars, except one example (v54 and V26 is not involved in this paper), agree well with Silbermann & Smith's (1995) values, our larger sample reveals an equal number of stars with positive (5) and negative (6) period changes. Barlai (1984) argued for equal numbers of positive and negative values for single mode RR Lyrae stars.
It seems that first overtones behave also in a different way in and stars, at least according to these samples.
Further observations at least for some of the stars in M 15 and similar investigations in other clusters are needed to clear up how general the conclusions of the present investigation are.
Theoretical investigations (effect of changes in the composition gradient (Cox, 1997) and variation in temperature and luminosity of both linear and nonlinear models (Kolláth, 1997)) for the explanation of the unambiguous case of v53 with the largest value of period change rates of different sign have been started.
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998