4. Period changes
The final results of the global fit of period changes are summarized in Table 4. In column 3 the new period values (1/ ( in Eq. 3.) are listed for the first moment of the observation, that is = 20724.652 for v17 - v61, = 24381.546 for v67 - v101, along with their errors. The rate of period change, and calculated in days/Myr and cycles/Myr are given in columns 5 and 6, respectively, along with their mean errors. The and values derived by Silbermann and Smith (1995) are also listed in Table 4. (columns 7 and 8). The present values are in good agreement with the rates (for v17,31,39,51 and 53) obtained by them but the present values are given with higher accuracy.
The newly derived periods were accepted for the final Fourier-phase diagrams. The phase changes of both fundamental and first overtone modes for 13 stars in M 15 are presented in Fig. 4. We should emphashize that the correlation between the curvature of the phase changes and period change is of the opposite sign from that seen in diagrams.
4.1. Comments on individual variables
Detailed inspection of the phase change diagrams in Fig. 4., taking into account the observational errors, suggests the following interpretations of individual stars. According to us, those stars could be seen as well-established double mode RR Lyrae stars where both phase change diagrams show simple, regular structure. Generally good agreement of our periods for the first overtone pulsation with those derived in previous studies of long term period changes has been obtained.
v17 Although the double mode nature is well-established, remarkable amount of correction in the period of both the fundamental (2% of the cycle/year alias) and the first overtone (6%) modes had to be applied. With the new value of periods the phase change diagrams are given with extremely low errors and the long-term behaviour is well-described by a negative parabola for the first overtone and positive parabola for the fundamental mode. This means period increase in the first overtone and period decrease in the fundamental mode. The period increase rate of the first overtone agrees with the value obtained by Silbermann and Smith (1995). The period decrease rate of the fundamental mode is one of the largest value in our sample.
v30 Only minor corrections (1% of the cycle/year alias value) in both frequencies were applied. The phase change diagrams show unique structure in our sample contradicting Silbermann and Smith (1995) result where "some scatter about a straight line but no significant evidence for period change" has been found. No doubt about the double mode nature but a sinusoidal fit is given for both modes with low error bars. The sinusoidal fit of the curves or phase diagrams (in single mode RR Lyrae stars) used to be a sign of the binary nature of the star. In our case, however, the two sinusoidal fits (practically with the same amplitude, and days for the first overtone and fundamental mode, respectively) are the same but are shifted with respect to each other, which rules out the binary nature. The cycle length of the phase change is 72 years. At this moment there is no definite explanation for v30.
v31 Minor corrections (3 and 1% for the first overtone and fundamental mode, respectively) were enough to get the accepted new periods. The double mode nature seems to be well-established but the overtone is dominant comparing to the fundamental mode. The amplitude ratio is rather high. The low amplitude of the fundamental mode is manifested through the higher error bars of the phase change curve of the fundamental mode. The first overtone's phase curve is fitted by a negative parabola with similar low value of period change rate as given by Silbermann and Smith (1995).
v39 Minor corrections (2% in both modes) were applied. We can confirm the previously published (Silbermann & Smith (1995), Purdue at al.(1995)) results concerning the double mode nature and the first overtone. The double mode nature is well-established and the evidence for a period decrease is weak. The phase change diagram is fitted by a positive parabola. The period decrease of the fundamental mode, in spite of the larger scatter around the curve, seems to be significant.
v41 This star is usually left out from the investigation of double mode stars. Silbermann & Smith (1995) defined only three epochs and the star was not included in their results. In the present investigation more numerous epochs are defined and the period decrease of the first overtone (correction was 1%) with rather large period change rate seems to be definitely significant. However, the Fourier phases are obtained with lower accuracy for the fundamental mode. No fit is given for the phase change of the fundamental mode (13% correction was applied). Any attempt failed to find a period value which gives Fourier phases with higher accuracy. The ill-defined solution could be attributed to the low amplitude of the fundamental mode.
v51 In the latest paper concerning the stars in M 15, Purdue at al. (1995) could not confirm the double mode nature during 1991-1992. The problem could be caused by the incorrect period determinations. Many trials were carried out to obtain the new values of the periods. Finally the first overtone period of Nemec (1985) and the +1 cycle/day alias of Jurcsik & Barlai's (1992) fundamental periods were accepted as proper starting values. A rather large correction (10%) of the first overtone frequency was applied. For the fundamental mode a smaller (1%) correction proved to be satisfactory. The phase change diagrams show regular structure, the double mode nature of v51 does not seem to be doubtful. The period change rate of the first overtone perfectly agrees with the value obtained by Silbermann and Smith (1995). The larger error bars of the fundamental mode phase change diagram could be the consequence of the relatively lower amplitude and/or the improper phase coverage in the groups. In spite of the larger error bars and scatter of points, a definite period increase with significant period change rate was derived.
v53 Although in most cases periods of the first overtone and fundamental mode obtained by the same authors were used, in this case the starting values of the first overtone given by Kovács et al. (1986) and that of the fundamental mode given by Purdue et al. (1995) were accepted. Minor correction (5%) for , but a larger one (7%) for were applied to get the new values. The Fourier phase diagram for both modes display clean, regular structure, that is, a positive parabola for the fundamental mode and a negative parabola for the first overtone with small scatters of the points along the parabola and small error bars of the derived phase in each group. The period changes of different sign for the first overtone and fundamental modes are even more pronounced than in v17. The period change rate of the first overtone perfectly agrees with the value obtained by Silbermann and Smith (1995). The period change rate of the fundamental mode is the largest significant value in our sample.
v54 Not too much improvement has been achieved for this star. Only minor corrections (0.5% and 2% for the first and second part of the phase change curve, respectively) for have been applied. Different trials, regarding some cycle/year aliases of the fundamental mode as starting values, were carried out. Finally the mean value of the +1 cycle/year alias of Purdue et al.'s (1995) period and the -1 cycle/year alias of Nemec's (1985) period was accepted as a starting value, but only a slightly better solution was found. (Corrections were 2% and 5% for the first and second part.) Concerning the large error bars of the Fourier phases, no fit is given for the phase change curve of the fundamental mode. The phase change of the first overtone is best fitted by two parabolas for the first and second parts of the data which are interpreted as different rates of period decrease. In between a days abrupt period increase occurred. The sharp break in the Fourier phase diagram (the abrupt period increase) rules out the possibility of a sinusoidal fit.
v58 and v61 As a consequence of less observation, only a few epoch could be derived for these stars. Although the Fourier phases are accurate enough, the period change rates do not seem to be significant. More observations are needed to get significant solutions.
v67 Minor corrections (4% for and 1% for ) were applied for Kovács at al. (1986)'s values. More numerous epochs, than derived by Silbermann and Smith (1995) give a definite Fourier phase diagram with high accuracy for both modes. The period change rates have different sign as in the cases of v17 and v53 but the values are rather low and do not seem to be significant. In any case, the regular phase change diagrams suggest that the new periods are well-defined values.
v96 Starting values were accepted as given by Nemec (1985). One of the largest correction (15%) has been applied for the first overtone. Although a minor correction (6%) was employed for the fundamental mode, too, the new value did not result in regular structure in the phase change diagram. Further observations are needed to find the precise period of the fundamental mode which has lower amplitude than the fundamental mode in the other stars. The period change of the first overtone is derived for the first time, since Silbermann and Smith (1995) did not include this star in their investigation. The first overtone period is decreasing with a significant period change rate.
v101 The starting values are also taken from Nemec (1985). This was one of the cases, where larger correction (6%) had to be applied for the fundamental mode than for the first overtone (3%). Although the fit is given to the phase diagram of the fundamental mode, the period change rate is at the limit of the significance. The first overtone shows a period decrease with significant period change rate.
4.2. General remarks
The phase changes are described by smooth curves over 80 years. A second order polynomial fit is satisfactory for most of the stars. The most striking result is the different period change behaviour of the fundamental and first overtone modes in rate and sign.
One can ask, whether it is possible to use the fit of the first overtone, to prewhiten the data and then to use the traditional method to determine the period changes in the fundamental mode. There are multiple reasons which make it impossible. First of all in real (noisy) data the maxima (or minima) are not well defined - and we use information only the neighbouring points around the extrema (contrary the Fourier phase method uses all the information from the light curve). The times of the maxima are also affected by the nonlinear coupling terms ( etc.), which are unknown. Those terms have only minor effects on the Fourier phase method, because the sine functions with well separated frequencies are almost orthogonal to each other even on the base of uneven sampling.
The regularities outlined here should give a new line of the investigation of pulsation and evolution connection. Comparisons with stars in other globular clusters should give dependence on the metallicity.
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998