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Astron. Astrophys. 322, 177-182 (1997)

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5. The main effect: an excess or deficit of objects

The majority of period data (Fig. 3) covers the [FORMULA] -range from about 2 to 45 µHz, the main commensurability effect being found at [FORMULA] Hz. However, because the [FORMULA] -function determines a particular test frequency [FORMULA] 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 [FORMULA] 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 [FORMULA] -function is determined by various contributions from the total set of frequency ratios [FORMULA]. If, indeed, a significant amount of objects exhibit a near-commensurability with frequency [FORMULA], the majority of deviations [FORMULA] for that frequency should be within the range 0.00 - 0.25; but, e.g., for the first overtone the substantially larger number of [FORMULA] -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 [FORMULA], or period P), for which [FORMULA] -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 [FORMULA] and [FORMULA], - 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).

[FIGURE] Fig. 3. The period distribution of CBS's with periods [FORMULA] d listed in five catalogues of binary stars ([FORMULA] ; periods P are expressed in days).

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 [FORMULA] 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 [FORMULA] -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 [FORMULA], [FORMULA], [FORMULA] etc. (According to (2) and (3), the sizes of the [FORMULA] -peaks in Fig. 2 vary approximately as the ratio [FORMULA] where [FORMULA] is the excess number contributing to a given "peak".)

Notice also that the [FORMULA] -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 [FORMULA], is equal to [FORMULA] (see expression (2) and Kotov 1986); consequently, the mean value of [FORMULA] is zero. Further, being normalized by the factor [FORMULA], it has a standard deviation of unity.

In Fig. 4 we plot the distributions of deviations [FORMULA] obtained for the [FORMULA] -spectrum (Fig. 2) at both frequencies, 104.160 and 52.080 µHz, and for the [FORMULA] -spectrum (Fig. 1) - at [FORMULA] Hz. We conclude that the basic effect is due to:

(a) a near-resonance (A effect) with frequency [FORMULA] - of an excess of [FORMULA] binaries ([FORMULA] 2%; see Fig. 4a);

(b) a lack of about 100 binaries with periods non-commensurate with [FORMULA] (the same A effect, with nearly identical relative amplitude, [FORMULA] %; see the same plot);

(c) a near-antiresonance (B -effect) of a roughly similar amount of binaries with respect to [FORMULA] (in fact, according to Fig. 4b, an excess of [FORMULA] binaries with [FORMULA] to 0.5, and a deficit of [FORMULA] binaries with [FORMULA]).

[FIGURE] Fig. 4. The distributions of deviations [FORMULA], obtained for the [FORMULA] -spectrum at frequencies a 104.160 µHz and b 52.080 µHz and c for the [FORMULA] -spectrum at [FORMULA] Hz (see text for description). The total number of sample frequencies [FORMULA] (for [FORMULA] d); the vertical bar indicates a typical [FORMULA] -uncertainty for each of ten bins ([FORMULA] for a and b, and [FORMULA] for c). To get the bottom plot "c", [FORMULA] -deviations of the plot "b" were transformed to deviations [FORMULA], then the new distribution was summed up with [FORMULA] -distribution of the plot "a". Bins with an excess and lack of [FORMULA] -deviations are shown by black and shaded areas, respectively.

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 [FORMULA] %). There is also a lack of about 200 binaries with [FORMULA] 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 [FORMULA] -spectrum plotted in Fig. 1. The highest peak in Fig. 2, with frequency [FORMULA] 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 [FORMULA] shown in Fig. 1.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998