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 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. (i = 1, 2, 3) for the same escaper of each of the families that
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 and The members of the family of escape orbits corresponding to these preselected r is associated with a given value of the parameters and For the above mentioned formulas (a)-(c) are applicable with sufficient accuracy.
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, G is the constant of gravity and are the units of mass, time and length and the subscript m refers to `model'. Consequently, the unit of time and the unit of velocity is 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 and In addition, it is expedient to introduce the symbols where is the distance between the binary. Moreover, as well as and 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, placed at the apices of an equilateral triangle at = 1 pc apart. The unit of time = 1.5 yr and the unit of velocity = 0.0655 Km/Sec. Here, = 1 and yr.
Model (II): The second model consists of three 40 Eridani-B- like stars with and The units of time yr and Km/Sec. Here, µ = 0.43, = 0.016 and
Model (III): The third model consists of three white dwarfs of solar masses (i.e. ) which placed at 1 pc distances. The radius of the participating bodies is following Chandrasekhar's estimate. The unit of time yr and the unit of velocity is Km/Sec. In this case much closer approaches are allowed without significant tidal effects. Here µ = 1, = 0.0068,
Model (IV): This model consists of three neutron stars of solar masses and placed at 1 pc distances. The radius of participating bodies is = 10 Km. The units of time yr and velocity = 0.0655 Km/Sec. Here, µ = 1, = 1.44 and
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 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 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. = 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 and escaper is 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 Since the bodies touch when For the purposes of this paper, we will select the following three values for
At this point some simple results are offered as follows:
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 R and A in astronomical units and in the fifth column, we have given magnitude of escape velocity determined from the formula mentioned in (ii) above and in the last column, we have mentioned magnitude of maximum relative velocity 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 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