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