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

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3. Equations of motion

Let there be three equal masses [FORMULA] each equal to unity occupy the vertices of an equilateral triangle [FORMULA]. At [FORMULA] the centre of mass, O, of the system is taken as the origin, x-axis parallel to one of the side [FORMULA] of the equilateral triangle [FORMULA] and y-axis lying in the plane of the triangle. We choose the unit of distance such that at [FORMULA] [FORMULA] [FORMULA] (Fig. 1).

[FIGURE] Fig. 1. Initial conditions.

The masses at [FORMULA] are subjected to small perturbing velocities [FORMULA] and [FORMULA] respectively such that [FORMULA] (i = 1, 2, 3). At [FORMULA], the position [FORMULA] and velocities [FORMULA] i = 1, 2, 3 of [FORMULA] are given by

[EQUATION]

The equations of motion of [FORMULA], having position vector [FORMULA] are given by

[EQUATION]

where [FORMULA] = 1, 2, 3 and G is the constant of gravitation.

By symmetry, the motion of [FORMULA] with zero perturbing velocity will take place along [FORMULA] and triple collision will occur at O. When there is no perturbation and [FORMULA] is the force between the ith and jth mass and the unit of time is so chosen that [FORMULA] then the equation of motion of each mass in terms of r, its distance from the origin is expressed as [FORMULA]

Taking [FORMULA] and solving, we obtain

[EQUATION]

This corresponds to the Kepler's equation. The colliding time is thus given by

[EQUATION]

The introduction of perturbing velocities break the symmetry and collision can be avoided. The distances between the masses are functions of [FORMULA] and [FORMULA] (i = 1, 2, 3).

Let [FORMULA] represent the lengths of the sides [FORMULA] respectively at time [FORMULA] In subsequent motion the values of the distances up to first order of t are

[EQUATION]

where

[FORMULA] [FORMULA] [FORMULA] [FORMULA] [FORMULA]

In the present study we require some more parameters at [FORMULA] which are given below:

(i) Moment of inertia I:

[EQUATION]

(ii) Total energy [FORMULA]:

Kinetic energy [FORMULA]

Potential energy [FORMULA]

Thus [FORMULA]

(iii) Angular momentum C :

[EQUATION]

(iv) Virial Coefficient [FORMULA]

[EQUATION]

Immediately after the motion starts, [FORMULA]
for [FORMULA]

From this inequality, it may be observed that

  1. the curve [FORMULA] is convex or concave from below,

  2. the total energy of the system is positive or negative,

  3. the virial coefficient of the system is [FORMULA] or [FORMULA] unity.

For small values of [FORMULA] (i = 1, 2, 3), the total energy [FORMULA] and [FORMULA] are negative and virial coefficient of the system is less than unity. This is evidently true for [FORMULA] and so the motion begins with contraction for all directions of the perturbing velocities in a plane.

Proceeding as in Szebehely (1974a), we can show that the minimum value of I, say [FORMULA], for initial collapse satisfies

[EQUATION]

It is also true for unequal masses as well. Moreover, this is more general than Szebehely's (1974b) result viz [FORMULA] which can be deduced from our result.

Since the distances of the escaper from the two-body which form a binary will go on increasing and eventually will be greater than the distance between the binary, the body which is likely to escape must be opposite to the shortest distance between the participating bodies. Keeping this criterion in view, it may be observed from the relative distances of the participating bodies that when low values of perturbing velocities [FORMULA] (i = 1, 2, 3) are introduced the initial symmetry gets destroyed and that there is a possibility of a body to escape which is opposite to the smallest side provided escape conditions are satisfied.

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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