5. The main effect: an excess or deficit of objects
The majority of period data (Fig. 3) covers the -range from about 2 to 45 µHz, the main commensurability effect being found at Hz. However, because the -function determines a particular test frequency which has a near-integer relationship to entire sample of data, one might expect to see many other harmonics (ranging, say, from 3-d to 50-th) of emerge in the resonance spectrum. The reason for the absence of those peaks (see Figs. 1 and 2) is the fact that at each test frequency the -function is determined by various contributions from the total set of frequency ratios . If, indeed, a significant amount of objects exhibit a near-commensurability with frequency , the majority of deviations for that frequency should be within the range 0.00 - 0.25; but, e.g., for the first overtone the substantially larger number of -deviations will spread up to 0.50, - so that the lowest harmonic might already be blurred out, and of course even more the higher harmonics (equally, in frequency , or period P), for which -values will be well randomized within the range 0.0 - 0.5. One should however note that some effect of harmonics might be still present in the real spectra, increasing thus the fluctuations of both and , - see below. This is similar to the influence of sidelobe structures on a power spectrum obtained unevenly-spaced data series; see, for instance, the discussion of the so-called quasi-persistency effect by Forbush et al. (1983).
When considering Fig. 2, one should keep in mind that a peak does not imply the presence of "excess" (or "lack") of objects at the corresponding frequency. For instance, the 104-µHz feature is not produced by an excess - say, by 100 - of binaries rotating with periods near 160 min, - as might be suggested from the vertical scale of Fig. 2. Any positive peak, according to (2), is determined as a matter of fact by a tendency of significant portion of objects to be near-commensurable, on the average, with a given period. The height of the -peak correlates with: (a) the relative number of near-commensurable objects and (b) the importance of that "tendency", i.e. with the percentage of data with deviations , , etc. (According to (2) and (3), the sizes of the -peaks in Fig. 2 vary approximately as the ratio where is the excess number contributing to a given "peak".)
Notice also that the -values are evenly distributed between 0.0 and 0.5 for pure noise; the mathematical expectation of the squares of the deviations, i.e. the mean of , is equal to (see expression (2) and Kotov 1986); consequently, the mean value of is zero. Further, being normalized by the factor , it has a standard deviation of unity.
In Fig. 4 we plot the distributions of deviations obtained for the -spectrum (Fig. 2) at both frequencies, 104.160 and 52.080 µHz, and for the -spectrum (Fig. 1) - at Hz. We conclude that the basic effect is due to:
(a) a near-resonance (A effect) with frequency - of an excess of binaries ( 2%; see Fig. 4a);
(b) a lack of about 100 binaries with periods non-commensurate with (the same A effect, with nearly identical relative amplitude, %; see the same plot);
(c) a near-antiresonance (B -effect) of a roughly similar amount of binaries with respect to (in fact, according to Fig. 4b, an excess of binaries with to 0.5, and a deficit of binaries with ).
The overall effect - the A and B resonances altogether, - as revealed by Fig. 4c, is due to an excess of about 200 binaries (when counting excesses of objects - in Fig. 4a,b - above the mean number of about 528 objects for each of the 10 bins). One can see, moreover, that about 200 extra binaries are tuned to A or B resonances within the relative limits of about 10% (or about 150 extra binaries - with an accuracy %). There is also a lack of about 200 binaries with with respect to both A and B resonances.
Three to five "peaks" in Fig. 2 have amplitudes comparable with that of the 104-µHz feature. An important fact is that none of those "peaks" has low-frequency satellites, unlike the 104-µHz feature which is "reproduced" by the remarkable 52-µHz negative satellite. This is also explicitly demonstrated by the -spectrum plotted in Fig. 1. The highest peak in Fig. 2, with frequency Hz which is located near the middle of two peaks of interest, "160 min" and "321 min", might be an artifact due to the method of computations; in any case it has no correspondance in the main spectrum shown in Fig. 1.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998