3. The method of analysis
A first method could be to check commensurability in the periods of binaries. According to Goldreich (1965), two frequencies, and (), are thought to be quasi-commensurate with each other if
where and are small integers and .
However in the present paper, following the Kotov & Koutchmy's (1985) approach, we shall compare rates of various objects with running frequency , which will vary within a given frequency range; i is the ordinal number of i -th object; where is the total amount of objects in the sample. If there is a frequency which shows a significant minimum of deviations of the ratios (or may be ) from an integer number, it will be near-commensurate with the rates of the total sample of objects.
To search for the resonance (or: near-commensurability) inside the sample of galactic binaries, we introduce the following commensurability function (CF; see Kotov 1986):
where if , and if ; and . By definition, the maximum of the -function corresponds to the best least-squares fit of the ratios - of pairs of frequencies - to integers. Kotov (1986) has shown that for a random sample of -s the standard deviation of CF equals one, and the -values themselves are normally distributed around zero.
Further, by analogy with usual power spectrum (PS) analysis, we define the commensurability spectrum
which, contrary to usual PS, takes into account also the sign of the CF: a positive (negative) value corresponds to a case of commensurability (non-commensurability).
One must realize that for an external GR (QGR) a binary in fact must appear as a two-fold object: (1) it may be in some sense perceived as a single "rigid" or "quasi-rigid" body (like dumb-bells), - especially in the cases of ellipsoidal, contact and semi-detached binaries (e.g., those of the W UMa type, with intense transfer of matter between components), or (2) just as a pair of separate stars (practically not interacting, bound only by gravitation force). Accordingly, we conjecture that there might be also a two-fold effect of a potential GW (QGW) of external origin (which has, say, a primary frequency ): (A) a simple resonance at frequencies most commensurate with , and (B) complementary resonances at a frequency and its integer harmonics.
The effects A and B however might be in opposite directions, if we consider the sign of . In the case of a "quasi-rigid" body (the A resonance) one expects an excess of objects with frequencies near-commensurate with (for indications for a similar situation see, e.g. Gough 1983 and Kotov & Koutchmy 1985). But in case of B resonance one expects a lack of binaries with and its integer harmonics, due to relatively rapid change of the binary period around this frequency(ies) caused by the gain of energy and angular momentum transferred by GR (or QGR). In other words, one may expect to find an excess of binaries with frequencies and , and also a lack of binaries with frequencies and , where Z is a positive integer.
To sum up both effects, we introduce a generalized CF:
with the corresponding spectrum
A maximum of at some frequency will indicate, on the average, the presence of a A type resonance (simple commensurability) at frequency and its integer harmonics, and simultaneously of a B type resonance (non-commensurability, i.e. a deficit of objects) at frequency and at its integer harmonics.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998