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

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7. Characteristics of the families

The following characteristics of the earlier mentioned families are observed for [FORMULA] [FORMULA]

Besides the observation of Sect. 6.1, we further observe that

  1. For all members of a family, the general shape of the orbits is the same so long so the escaper remains the same.

  2. We know that

    [EQUATION]

    When a body escapes, its distances from the other two bodies tend to [FORMULA] and F varies according to [FORMULA] where i and j do not take up the values corresponding to the suffix of the escaper. This also means that F is minimum or maximum according as the distance between the binary is maximum or minimum. Thus we conclude that the value of F is governed by one single close binary approach after [FORMULA] is attained and not before it.

  3. When the perturbing velocities have the same magnitude and direction, by symmetry the relative distances between the participating bodies remain equal. But if directions of the perturbing velocities are not equal (keeping the magnitude of the velocities equal), we are able to say which body amongst them is going to escape with the formation of a binary.

  4. The quantity [FORMULA] (the total mass is three), which is used as the radius of gyration by Birkhoff (1927) and Sundman (1912) instead of moment of inertia, always hold the following inequalities


    [TABLE]

  5. The magnitude of velocity of the latecomer (escaper) after the time of first close approach increases and the escaper moves towards the centre of mass.

  6. The latecomer (escaper) must pass close to the centre of mass after the first close approach.

  7. In each family, the body for which the perturbing velocity is varied never escapes.

  8. The semimajor axis [FORMULA] and its eccentricity e for case (I) (Sect. 4) of the first family with their sub- cases are shown in Figs. 4a and 4b.

  9. Proceeding as in Szebehely (1974a), it can be shown that

    [EQUATION]

    or [FORMULA] for all families.
    The relation is in good agreement with the numerical results. We have shown the value of [FORMULA] for both the sub cases of the first family for case (I) (Sect. 4) in Figs. 5a and 5b.

    It may be noted from the above Figs. 4a and 4b that for the family 1(a), when [FORMULA] varies from [FORMULA] to [FORMULA] and other parameters are fixed, the semimajor axis [FORMULA] decreases as [FORMULA] increases up to a certain value and then a rapid fall. Further, eccentricity e decreases slowly with [FORMULA] up to a certain value and then there is sharp decrease and after that it increases rapidly. And for the family 1(b) when [FORMULA] varies from [FORMULA] to [FORMULA] and other parameters are kept fixed as above, the semimajor axis [FORMULA] and its eccentricity e fluctuate. These trends are the same for all other families. Further, it may also be noted in Figs. 5a and 5b that for the first family 1(a) the escape velocity [FORMULA] decreases slowly as [FORMULA] increases up to a certain value and after that it increases rapidly. And for the family 1(b), escape velocity [FORMULA] decreases as [FORMULA] increases up to a certain value after that it increases rapidly.

  10. The minimum moment of inertia [FORMULA] and the difference [FORMULA] between [FORMULA] and [FORMULA] of the families are characterised. For the family 1(a) for case (I) (Sect. 4), [FORMULA] decreases slowly up to a certain value as [FORMULA] increases after that it decreases sharply and then increases rapidly where as [FORMULA] increases slowly up to a certain value with [FORMULA] after that it increases rapidly (Fig. 6a). For the family 1(b) [FORMULA] and [FORMULA] fluctuate (Fig. 6b). The trends for [FORMULA] and [FORMULA] are the same for all other families.

[FIGURE] Fig. 4a. Semimajor axis [FORMULA] and eccentricity e Vs [FORMULA] [FORMULA]; [FORMULA], [FORMULA]; [FORMULA], [FORMULA]

[FIGURE] Fig. 4b. Semimajor axis [FORMULA] and eccentricity e Vs [FORMULA] [FORMULA]; [FORMULA], [FORMULA]; [FORMULA], [FORMULA]

[FIGURE] Fig. 5a. Escape velocity [FORMULA] Vs [FORMULA]; [FORMULA], [FORMULA]; [FORMULA], [FORMULA]

[FIGURE] Fig. 5b. Escape velocity [FORMULA] Vs [FORMULA]; [FORMULA], [FORMULA]; [FORMULA], [FORMULA]

[FIGURE] Fig. 6a. Time to triple close approach [FORMULA] and minimum moment of inertia [FORMULA] Vs [FORMULA]; [FORMULA], [FORMULA]; [FORMULA], [FORMULA]

[FIGURE] Fig. 6b. Time to triple close approach [FORMULA] and minimum moment of inertia [FORMULA] Vs [FORMULA]; [FORMULA], [FORMULA]; [FORMULA], [FORMULA]

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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