SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

6. Numerical analysis

The numerical integration is performed by two different computer programmes through Unix System to obtain consistent and reliable numerical results. We have chosen an upper bound for 12 places of decimal for the local truncation error in both the programmes.

The first numerical integration is performed by using Runge-Kutta-Fehlberg 7(8) method. The second programme is developed by Sverre Aarseth, IOA, Cambridge, with the Aarseth & Zare (1974) regularization applied simultaneously to the two smallest distances at all times. The programme is able to handle the values of [FORMULA] (i = 1, 2, 3) up to the order of [FORMULA]

6.1. Actual regions of escape

As the purpose of this paper is to investigate triple close approaches with systematic regularity of escape with the formation of a binary, so it is restricted to low velocities of the participating bodies. It has been observed that keeping two velocities fixed, the maximum magnitude of the third velocity for which the escape occur with the formation of a binary depends upon its direction. It is denoted by [FORMULA] (i = 1, 2, 3) (Fig. 2).

[FIGURE] Fig. 2. Maximum initial velocity to escape with the formation of a binary [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA]

It is observed that the values of [FORMULA] fluctuates between maximum and minimum values. When the magnitude of the velocity is greater than [FORMULA] the escape changes into ejection or interplay. Here, escape means that escape occurs simultaneously with the formation of a binary whereas in the case of interplay, the bodies performed repeated close approaches and in the case of ejection the two bodies form a binary and the third body ejected with elliptic relative velocity (Szebehely 1973). The trend remains the same in all other cases.

Now we define the following families:

1. [FORMULA] varies and [FORMULA] fixed,

2. [FORMULA] varies and [FORMULA] fixed,

3. [FORMULA] varies and [FORMULA] fixed.

There are two sub-cases in each of these three families,

1(a) varying magnitude [FORMULA] and keeping direction [FORMULA] fixed,

(b) varying direction [FORMULA] and keeping magnitude [FORMULA] fixed.

Similarly for the cases (2) and (3).

We have studied these families with perturbing velocities varying from [FORMULA] to [FORMULA] and for all values of [FORMULA] i.e. [FORMULA] (i = 1, 2, 3).

In actual experiment escape with the formation of a binary does occur according to our analysis in Sect. 4.

As our main aim is to study in detail the effect of the perturbing velocities on various parameters, we have taken typical representative member of the cases 1(a) and 1(b) of the first family. For case 1(a), we have taken [FORMULA] [FORMULA] and [FORMULA] varying between [FORMULA] and [FORMULA] with the grid of [FORMULA] having directions [FORMULA] [FORMULA] and [FORMULA] (Table 1) and for case 1(b) [FORMULA] [FORMULA] [FORMULA] having directions [FORMULA] [FORMULA] and [FORMULA] lying between 0 and [FORMULA] at an interval of [FORMULA] (Table 2).


[TABLE]

Table 1. Timings [FORMULA] & [FORMULA] & [FORMULA] [FORMULA] [FORMULA] and escaper corresponding to [FORMULA] [FORMULA]



[TABLE]

Table 2. Timings [FORMULA] & [FORMULA] & [FORMULA] [FORMULA] [FORMULA] and escaper corresponding to [FORMULA] [FORMULA]


In these tables, in the column of time, there are two values against each [FORMULA] for Table 1 and each angle [FORMULA] for Table 2. The first value gives the time [FORMULA] for the first close approach and the second [FORMULA] at the time of second close approach. Here, first close approach means the first minimum relative distance between the participating bodies and the second close approach means the smallest relative distance between the participating bodies when minimum moment of inertia is attained. Corresponding to these timings the other columns in the two tables give the values of the moment of inertia [FORMULA] and [FORMULA] the distances [FORMULA] from the centre of mass, the relative distances [FORMULA] the magnitude of the velocities [FORMULA] and the relative velocities [FORMULA] of the three participating bodies. We have also calculated absolute potential [FORMULA] virial coefficient [FORMULA] and escape probability [FORMULA] that is linked with the behaviour of escaper according to Agekian's et al. (1969) and Szebehely (1974b). The escaping body is mentioned in the last column.

In actual experiments we have observed that

  1. two close approaches occur before the formation of a binary.

  2. minimum smallest relative distance of the smallest relative distance among the relative distances of the participating bodies occur at the time of first close approach.

  3. [FORMULA] occurs slightly latter than the first close approach.

  4. the body which escapes is opposite to the minimum relative distance [FORMULA] corresponding to the first close approach, though the actual escape occurs after [FORMULA] is attained. The distances of the escaper from the two bodies forming the binary go on increasing to infinity as t goes to infinity.

  5. the two bodies, amongst the participating bodies having the first close approach, are different from the two bodies which have the second close approach.

  6. the values of virial coefficient [FORMULA] F and [FORMULA] varies for each family. For case 1(a), of the first family these vary up to a certain value of [FORMULA] after that they remain constant, and for case 1(b) they vary with [FORMULA]

Similarly, we can deal with the other two families.

6.2. Orbits near triple close approaches

The orbits near triple close approaches for a typical representative member of a family (Case (I), Sect. 4) have been shown in Fig. 3.

[FIGURE] Fig. 3. Orbits near triple close approaches [FORMULA], [FORMULA]; [FORMULA], [FORMULA]; [FORMULA], [FORMULA]

The time t [also the transformed time [FORMULA] corresponding to the points of the above figure are listed in Table 3. The first close approach occurs at the point B and the second close approach (when [FORMULA] occurs) at the point C.


[TABLE]

Table 3. Time t and transformed time [FORMULA] corresponding to points on the orbits in Fig. 3


We observe that at the time of first close approach, the minimum moment of inertia does not occur. The reason is that when asymmetric initial conditions are introduced, just then all the three bodies begin their motion with a contraction towards the centre of mass and an interesting feature occurs between two bodies. These two bodies experience a first close approach while the third body is delayed. At the time of this first close approach, the moment of inertia is not minimum, since the delayed third body called latecomer for all members of the family still moves towards the centre of mass along with another body. The latecomer comes closest to the centre of mass with the other two and experiences a second close approach. After the minimum moment of inertia is attained, the latecomer at the time of first close approach is always the escaping or ejected body and the other two that experience close approach at this time form a binary.

We have carried out experiments for a large values of [FORMULA] where [FORMULA] lies between 0 and [FORMULA] and in each family, we see how a condition of complete collapse may be perturbed to obtain well-established families of asymmetric triple close approaches with systematic regularity of escape of the third body with the formation of a binary. Thus, we see that infinite escapes occur with the formation of a binary which indicate that the conjecture of Szebehely (1977), viz. "The measure of escaping orbits is significantly higher than the measure of stable orbits", is likely to be true.

6.3. Comparison with Agekian's escape probability

Agekian's et al. (1969) have defined the probability of escape

[EQUATION]

(Strictly speaking [FORMULA] is not a probability as it is already [FORMULA]1).

The same definition is used by Szebehely (1974b). Both of them linked the behaviour of [FORMULA] with the escape and the formation of a binary. Since [FORMULA] the total energy is fixed, so it depends on the absolute value of F which varies with [FORMULA] The details may be seen in Tables 1 and 2, where the first and second rows of each [FORMULA] (or [FORMULA]) correspond to the first and second close approaches respectively. The point B (Table 3, see also Fig. 3) which corresponds to the first close approach, the absolute value of the potential F is maximum. At the point C (Table 3, see also Fig. 3) which corresponds to the second close approach and at this point the absolute value of the potential F is another maximum but smaller than at the point B. Consequently [FORMULA] i.e. [FORMULA] And therefore, the minimum moment of inertia is not associated with the maximum absolute value of the potential. This fact agrees with Agekian's et al. (1969) observations that F may be governed by one single close binary approach while I is governed by the closeness of all three bodies which is indeed for escape and which is an indication of a forthcoming escape. But on the other hand, we have also some cases where minimum moment of inertia [FORMULA] is associated with maximum absolute value of the potential F and in such case [FORMULA] i.e. [FORMULA] For example, when case (II) (Sect. 4; [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA]) is taken, we have [FORMULA] and [FORMULA] It means [FORMULA] Though this does not indicate the forthcoming escape but in actual experiment escape does occur with the formation of a binary.

Thus we conclude that the value of [FORMULA] that links the behaviour of escape according to Agekian's et al. (1969) does not necessarily indicate the forthcoming escape for all perturbing velocities. Hence our result is in contrast with Agekian's et al. Here the reason is that the value of F is governed by one single close binary approach after [FORMULA] is attained and not before it, whereas [FORMULA] is governed by the closeness of all the three bodies. So we cannot connect F with [FORMULA]

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
helpdesk.link@springer.de