## 6. The test using random numbersTo 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
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
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.
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
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |