## 8. ApplicationsEarly references regarding individual orbits or statistically established families exist in the literature (Agekian & Anosova 1967, Szebehely & Peters 1967 and others). Nahon (1973), Waldvogel (1977), Szebehely (1974a,b, 1979), Chandra & Bhatnagar (1998a) have also investigated dynamical methods that is closely related to this problem. The present study is based on a family rather on individual orbits. So it allows the continuous adjustment and matching of physical parameters. For physical consideration, we shall consider the relative distances of the bodies at the time which has the minimum smallest relative distance between the first and the second close approaches. To simplify the notation in the sequel, we denote asymptotic value of the semimajor axis as the relative distances and of the participating bodies as respectively at the time of first or second close approach (which has the minimum smallest relative distance). The magnitude of relative velocity between the bodies at the time of having first close approach as and the hyperbolic escape velocity of the escaper as In actual experiment, we have found in the range of
i.e.
( -
(a) the minimum smallest relative distance between the bodies at the time of first or second close approach (which has the minimum smallest relative distance) is directly proportional to the semimajor axis of the binary ` *a*' i.e. = const. = (say). Here*r*is the minimum of -
(b) = 2, -
(c) = const. = (say).
These results mean that if in the non-dimensional system used, a
minimum relative distance The system reaches minimum size or rather minimum moment of inertia in approximately = 0.6413 non- dimensional time-units. All results of numerical aspects of triple close approaches are
presented in non-dimensional form to facilitate any applications. The
non-dimensional parameter is in the
numerical investigation. Here,
Model (V) of Szebehely (1979) involving galaxies cannot be taken as galaxies are not point masses. Here, our main aim is to find the principal characteristics of the
models. Such characteristics are the velocity of escape At this point some simple results are offered as follows: -
the semimajor axis of the binary formed is -
the velocity of escape is -
the magnitude of highest relative velocity i.e.
The results regarding the four models are given in Table 4.
The first column gives the model. The second column gives various
values of taken, as mentioned
above. The third and fourth columns give
From the above table, we observe that the highest values of and among all the models occur in the model (IV) that is for the model of three neutron stars. The highest value of corresponds to approximately 19% of the velocity of light. © European Southern Observatory (ESO) 1999 Online publication: May 21, 1999 |