3. Fourier phase method
3.1. New period values
Since the observational data are seasonal or even several-year-long gaps could be found, the chance of miscounting the number of cycles increases; that is, the alias pattern of the frequency spectrum is more complicated. Although many careful analyses, using a part of the available data and different techniques (PDM, Fourier), have been made during the past years (Smith & Sandage, 1981; Cox et al., 1983; Nemec, 1985; Kovács et al., 1986; Clement & Walker, 1991; Jurcsik & Barlai, 1992; Silbermann & Smith, 1995; Purdue et al., 1995), there is still dispute paper by paper concerning the 1 or 2 cycles/year aliases of the lower amplitude fundamental mode, for example in v39, 30 and 31. In critical cases some extra criteria like symmetry arguments or aliases with physically realistic primary/secondary period ratios are used. At the same time, any kind of period changes could only be obtained if the precise value of the pulsation period is well-known, otherwise the period change is masked by the effect of incorrect value of the period.
The traditional phase diagrams or curves based on all of the available observational data can give a very accurate value and even the change of the dominant (first overtone) period. However, there is no possibility for the investigation of the lower amplitude fundamental mode. The combination of the traditional phase or diagram and the Fourier technique, which we call the Fourier-phase diagram method, can give not only very accurate values but the period changes of both excited modes in stars.
Since the effect of an incorrect frequency or a change in the frequency accumulates slowly, the data were devided into groups of different length in time according to the number of observations in a group, not more than two-three years were added together. Although stars in M 15 were more or less homogeneously observed the duration of data segments is not the same for each star, since the phase solution is sensitive to the data coverage (different for stars with different periods). In a critical case a longer data segment was accepted.
In finding the new period values the program PERIOD (Breger, 1990) was used with an option of fixing the frequencies and amplitudes, only phases (given in 3.2) were considered as free parameters. The reduced number of free parameters allowed us to use more sporadic observations, too. The starting values of the periods were accepted from the paper of Kovács et al. (1986), in some cases that of Nemec (1985) or Purdue et al. (1995). Amplitudes were fixed as it is described at the global fit in 3.2.
Although for checking the stability of the phase solution, 5 different sets of linear combinations have been studied, the linear combination has been applied in the final phase diagrams.
A slightly incorrect (not precise enough) starting value of the period gives slightly different values for the consecutive groups. If the starting periods are precise enough comparing to the 1 cycle/year aliases, the differences accumulate to one cycle over the interval of some groups, lying along a segment of a straight line. After modulation by 2 the consecutive phases are situated on parallel segments of straight lines. The phase diagram in this stage looks like a saw-tooth function. During the demodulation the parallel segments of straight lines are shifted to a single straight line. If the uncertainty of the starting value of the period is comparable to the 1 cycle/year aliases, it results in a 2 phase change over the length of a group, there is no way to get the correct period (miscounting of cycles). Another alias as a starting value is suggested.
The plot of phases ( 2 ,4 ...) versus the mean time of the observations in a group gives the Fourier phase diagram. The frequencies had to be increased or decreased comparing to the starting values. The value of frequency correction is given by the slope of the straight line which is very definite.
In Fig. 2. the Fourier-phase diagram of a remarkable star v67 is plotted for the first overtone and fundamental modes. The star v67 was chosen as an example where the amplitude of the fundamental mode is not essentially lower than that of the first overtone and previously there was no definite result for this star, nor for .
In Fig. 3. the folded light curves of v67 according to the phase of the first overtone and the fundamental mode, respectively, could be seen. At the left side the starting values are used, while at the right side the new values have been used. The improvement of the folded curves with the new frequencies is remarkable.
The accepted frequency corrections for the stars in M 15 according to the Fourier-phase diagram method are given in Table 3. The finally accepted new values applied in the final fit, are presented in column (3) of Table 4.
Table 3. Frequency corrections applied to the previously published values
Table 4. Period changes of stars in M 15
3.2. The global fit of period changes
The changes of the Fourier phases were determined by a nonlinear least square fit to the light curve () i.e. by minimizing the following function:
where the phase term is given by
The amplitudes () were held constant at values determined by a fit to the whole data set, while the zero shifts () were calculated for the data segments independently to minimize the errors due to the different observations. Only the parameters were taken as unknown values during the nonlinear iteration.
The frequency correction and variation is determined by the parameters:
The errors of the periods and the rates of period variations ( and ) were calculated by a Monte-Carlo simulation. Different realizations of Gaussian noise were added to the observed magnitudes and then the fit was recalculated. Then and was obtained from the standard deviation of the resultant and factors. The standard deviation of the added noise was selected according to the observational noise ().
We tested the method on an artificial data set given by our fit to the v53 light curve. The times of the observations were used in the test signal. With different amount of added noise () in the test signal the dependence of the errors and on the observational noise was determined. As expected, we found linear relations: , , and .
The amplitudes () are held time independent in our calculations, so it is important to test how the results depend on amplitude variations. For this test, each segment of the light curve was multiplied by different random factors , where z is a Gaussian noise with standard deviation . We again found linear dependences: , , and .
Even for a high value of the amplitude fluctuation the errors of the period and period change rates remain small, i.e. the V and the B light curves can be mixed, without altering the results significantly. Although the displayed values are valid only for the data distribution of the V53 light curve, they show the general tendencies of error propagation.
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998