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

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4. Possible regions of escape

From relative distances of the participating bodies for small value of t, it may be possible to point out the body which is likely to escape with the formation of a binary. We may have [FORMULA] and [FORMULA] When we combine each of [FORMULA] with each of [FORMULA] for small values of [FORMULA] we get the following cases.

Case (I) : when [FORMULA] and [FORMULA].

It means [FORMULA] is the smallest and therefore [FORMULA] may escape and [FORMULA] form a binary.

Case (II) : when [FORMULA] and [FORMULA].

Thus we have [FORMULA] may escape and [FORMULA] form a binary.

Case (III) : when [FORMULA] and [FORMULA].

We cannot take a decision as it is not possible to point out the smallest distance.

Case (IV) : when [FORMULA] and [FORMULA].

Thus we have [FORMULA] may escape and [FORMULA] form a binary.

Case (V) : when [FORMULA] and [FORMULA].

Now there are three possibilities according as [FORMULA]

  1. (a) When [FORMULA], we have [FORMULA] may escape and [FORMULA] form a binary.

  2. (b) When [FORMULA], it means [FORMULA] is smallest and therefore [FORMULA] may escape and [FORMULA] form a binary.

  3. (c) When [FORMULA]. We can not take a decision due to equality sign.

Case (VI) : when [FORMULA] and [FORMULA].

Thus we get [FORMULA] may escape and [FORMULA] form a binary.

Case (VII) : when [FORMULA] and [FORMULA].

Here, we can not take a decision due to equality sign.

Case (VIII) : when [FORMULA] and [FORMULA].

It means [FORMULA] may escape and [FORMULA] form a binary.

Case (IX) : when [FORMULA] and [FORMULA].

Here, we cannot take a decision due to equality sign.

The above results may change for higher values of t corresponding to certain directions of the perturbing velocities given to the participating bodies but escaper must be opposite to the smallest relative distance of the participating bodies.

From the above analysis, we conclude that out of the 9 cases, we may be able to take a decision only in 6 cases.

In the literature, many escape conditions exist. By Sundman (1912), it is sufficient for escape that

[EQUATION]


[TABLE]

According to the above escape condition, [FORMULA] form a binary and [FORMULA] escapes. It has to be modified according to the escaper.

In our case, the above condition is satisfied and escape does occur for sufficiently small values of [FORMULA] (i = 1, 2, 3). This follows from the fact that as [FORMULA] The asymmetric triple close approaches after reaching [FORMULA] generates sufficiently large values of I and [FORMULA] for escape with the formation of a binary. After attaining of [FORMULA] one of the three bodies that is opposite to the shortest distance escapes and the remaining two form a binary.

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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