4 Multiple frequency determination

4.1 Multi-frequency analysis

Among the data sets listed in Table 2, VZ2a, VZ2b, VZ2c have few points and VZ4a, VZ4b, VZ4c, although they include more measurements, show mainly increasing segments with few decreasing parts (see Fig. 7), which are not available to deep multiple frequency analyses due to the artificial sampling effect. So the data set VZ1 and VZ3 are used to do Fourier transformation.

The multiple-frequency analyses were performed with the computer program MFA (Hao 1991). After the non-equal spacing Fourier transformation (Deeming 1975) was done, the highest power peak was chosen automatically, or the peak was chosen by hand, then both linear and nonlinear least square fits (Hao 1991) were used to determine the best frequency and corresponding amplitude and phase. The data were then prewhitened with that frequency and the process repeated to search for the next frequency automatically. Step by step, the best frequencies and fit results were obtained. To check the reliability of the results, the theoretical light curves were calculated using the following formula,

$$m(t)=m_0+\sum _{i}A_i\cos [2\pi f_i(t-t_0)+\phi _i]$$

where t₀ is the beginning time; A_i, the amplitude; and $\phi _i$,the phase. The result of the analysis provides the multiple-frequency solution and the residual, as well as various power spectra where the best multiple frequency fits have been prewhitened.

The analysis results on data set VZ1 and VZ3 are listed in Tables 3 and 4. Figures 2 and  3 show the spectral windows and power spectra of VZ1 and VZ3, respectively. In Figs. 4 and  5, the observation points and fit light curves of VZ1 and VZ3 are shown respectively. The residuals of VZ1 and VZ3 after 6 frequency fittings are 0.021 mag and 0.027 mag, respectively.

  

Table 3: Results of the Fourier analysis for VZ1

  

Table 4: Results of the Fourier analysis for VZ3

  

Figure 2: Spectral window and power spectra of the data set VZ1. Multiple pulsation frequencies are extracted one by one

  

Figure 3: Spectral window and power spectra of the data set VZ3. Multiple pulsation frequencies are extracted one by one

  

Figure 4: Observation points and fit curves of VZ1. The abscissa is HJD 2449000.0+, while the ordinate is $\triangle$V (mag)

  

Figure 5: Observation points and fit curves of VZ3. The abscissa is HJD 2436000.0+, while the ordinate is $\triangle$V (mag)

4.2 Fit to other data sets

As mentioned above, the other data sets are not suitable to do independent multiple-frequency analysis. So the 6 frequencies obtained from the analysis for VZ1 (see Table 3) are used to do fitting on data set VZ2a, VZ2b, VZ2c and VZ4a, VZ4b, VZ4c. Table 5 lists the fitting results. Figures 6 and  7 show the observation points and corresponding fitting light curves for VZ2a and VZ4a, respectively. The agreement between the observations and the fitting curves for VZ2b, VZ2c and VZ4b, VZ4c, which are not displayed in this paper, is as good as those for VZ2a and VZ4a, respectively.

  

Figure 6: Observation points and fit curves of VZ2a. The abscissa is HJD 2430000.0+, while the ordinate is $\triangle$V (mag)

  

Figure 7: Observation points and fit curves of VZ4a. The abscissa is HJD 2433600.0+, while the ordinate is $\triangle$V (mag)

  

Table 5: Fitting results for VZ2a, VZ2b, VZ2c and VZ4a, VZ4b, VZ4c with the 6 frequencies