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Astron. Astrophys. 346, 652-662 (1999)

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8. Applications

Early 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 [FORMULA] as [FORMULA] the relative distances [FORMULA] and [FORMULA] of the participating bodies as [FORMULA] 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 [FORMULA] and the hyperbolic escape velocity [FORMULA] of the escaper as [FORMULA]

In actual experiment, we have found in the range of [FORMULA] i.e. [FORMULA] (i = 1, 2, 3) for the same escaper of each of the families that

  1. (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. [FORMULA] = const. = [FORMULA] (say). Here r is the minimum of [FORMULA]

  2. (b) [FORMULA] = 2,

  3. (c) [FORMULA] = const. = [FORMULA] (say).

These results mean that if in the non-dimensional system used, a minimum relative distance r at the time of first close approach is selected, we may compute [FORMULA] and [FORMULA] The members of the family of escape orbits corresponding to these preselected r is associated with a given value of the parameters [FORMULA] and [FORMULA] For [FORMULA] the above mentioned formulas (a)-(c) are applicable with sufficient accuracy.

The system reaches minimum size or rather minimum moment of inertia in approximately [FORMULA] = 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 [FORMULA] in the numerical investigation. Here, G is the constant of gravity and [FORMULA] are the units of mass, time and length and the subscript m refers to `model'. Consequently, the unit of time [FORMULA] and the unit of velocity is [FORMULA] Here, note that in the following discussion lower case letters will denote the dimensionless quantities, while capitals will be reserved for dimensional quantities. The relations between these symbols are [FORMULA] [FORMULA] and [FORMULA] In addition, it is expedient to introduce the symbols [FORMULA] where [FORMULA] is the distance between the binary. Moreover, as well as [FORMULA] and [FORMULA] The theory so developed can be applied to any desired model without difficulties. We have, therefore, considered the following four astronomical models as in Szebehely (1979).

Model (I): The first model consists of three Sun-like stars. For characteristic units, [FORMULA] placed at the apices of an equilateral triangle at [FORMULA] = 1 pc apart. The unit of time [FORMULA] = 1.5 [FORMULA] yr and the unit of velocity [FORMULA] = 0.0655 Km/Sec. Here, [FORMULA] = 1 and [FORMULA] yr.

Model (II): The second model consists of three 40 Eridani-B- like stars with [FORMULA] and [FORMULA] The units of time [FORMULA] yr and [FORMULA] Km/Sec. Here, µ = 0.43, [FORMULA] = 0.016 and [FORMULA]

Model (III): The third model consists of three white dwarfs of solar masses (i.e. [FORMULA]) which placed at 1 pc distances. The radius of the participating bodies is [FORMULA] following Chandrasekhar's estimate. The unit of time [FORMULA] yr and the unit of velocity is [FORMULA] Km/Sec. In this case much closer approaches are allowed without significant tidal effects. Here µ = 1, [FORMULA] = 0.0068, [FORMULA]

Model (IV): This model consists of three neutron stars of solar masses and placed at 1 pc distances. The radius of participating bodies is [FORMULA] = 10 Km. The units of time [FORMULA] yr and velocity [FORMULA] = 0.0655 Km/Sec. Here, µ = 1, [FORMULA] = 1.44 [FORMULA] and [FORMULA]

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 v of the escaper, semi-major axis a of the binary formed, its eccentricity [FORMULA] and the time necessary to reach maximum contraction. The time necessary from the beginning of the motion to the occurrence of the escape in all cases is approximately [FORMULA] time units. Therefore, the escape times are determined by the time units applicable to the various cases. Here, we have taken typical representative member of both the sub-cases 1(a) and 1(b) of the first family. For this, we have observed from the Tables 1 and 2 that the values of earlier mentioned formulas (a)-(c) are constant (i.e. [FORMULA] = 11) for the same escaper for both the sub-cases of the first family as mentioned earlier. Here, smallest relative distance at the time having the first close approach is [FORMULA] and escaper is [FORMULA] The method of selection of the proper member of the families are based on the choice of minimum distance from physical considerations. This minimum distance R during a realistic, physically possible motion must be considerably longer than [FORMULA] Since [FORMULA] the bodies touch when [FORMULA] For the purposes of this paper, we will select the following three values for [FORMULA]


At this point some simple results are offered as follows:

  1. the semimajor axis of the binary formed is [FORMULA]

  2. the velocity of escape is


  3. the magnitude of highest relative velocity i.e. [FORMULA]

The results regarding the four models are given in Table 4. The first column gives the model. The second column gives various values of [FORMULA] taken, as mentioned above. The third and fourth columns give R and A in astronomical units and in the fifth column, we have given magnitude of escape velocity [FORMULA] determined from the formula mentioned in (ii) above and in the last column, we have mentioned magnitude of maximum relative velocity [FORMULA] by using formula given in (iii) above.


Table 4. Applications to models of three-body systems

From the above table, we observe that the highest values of [FORMULA] and [FORMULA] among all the models occur in the model (IV) that is for the model of three neutron stars. The highest value of [FORMULA] corresponds to approximately 19% of the velocity of light.

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Online publication: May 21, 1999