6. The test using random numbers
To make the above arguments more convincing, we repeated computations using 5845 random numbers - random "orbital periods" , with a -distribution identical with the real one (Fig. 3). The spectrum computed for 5280 random periods d is shown in Fig. 5 where no significant peak is present. In particular, we do not see any noticeable feature at the frequencies of interest, near 52 and 104 µHz.
The generalized commensurability spectrum computed for those 5280 random numbers, is shown in Fig. 6; there is also no significant positive peak exceeding . Some negative peaks - in particular that at Hz (with h), see Figs. 1 and 6, - are thought to result from inhomogeneities of the P -distribution (Fig. 3) associated perhaps with observational selection effects or physical properties of binaries. The fact that those peaks are negative, exclude a resonance effect at the corresponding frequencies. It is interesting that the negative peak Hz emerges in both -spectra shown in Fig. 1 (real periods ) and Fig. 6 (fictitious periods ). This must be attributed plausibly to relatively sharp variations in the period distribution (Fig. 3) which are identical for both samples.
The formal significance of a peak in the or spectrum, expressed in units of 's of a normal distribution, is , where is the peak amplitude of the "quadratic" spectrum or , and ie the standard deviation of the corresponding function, or (see expressions (2)- (5)). For pure noise, the standard deviation of both and is 1.0 (Kotov 1986). The slight overlap between catalogues increases and therefore leads to a decrease of the ratio and thus to a decrease of . To overcome this problem, we estimate empirical -values for each spectrum, and being considered in sufficiently wide frequency ranges; the results are given in Table 1.
Table 1. The empirical -values obtained for 5280 sample periods d.
As expected, the empirical -value of random data in the spectrum, agrees well with the theoretical value 1.00. The small increase of 's for other three type of data might be easily explained by (a) a quasi-persistency of - and -spectra due to method of computation (via the potential presence of harmonics of "peaks") and (b) the overlap between catalogues (in the case of real data). Taking into account the entries of Table 1, the confidence level of the main -resonance in Fig. 1 is decreased to .
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998