## 3. Equations of motionLet there be three equal masses
each equal to unity occupy the vertices of an equilateral triangle
. At
the centre of mass,
The masses at are subjected to
small perturbing velocities and
respectively such that
( The equations of motion of , having position vector are given by where = 1, 2, 3 and By symmetry, the motion of with
zero perturbing velocity will take place along
and triple collision will occur at
Taking and solving, we obtain This corresponds to the Kepler's equation. The colliding time is thus given by The introduction of perturbing velocities break the symmetry and
collision can be avoided. The distances between the masses are
functions of and
( Let represent the lengths of the
sides respectively at time
In subsequent motion the values of
the distances up to first order of where
In the present study we require some more parameters at which are given below: (i) Moment of inertia I: (ii) Total energy : Kinetic energy Potential energy Thus (iii) Angular momentum (iv) Virial Coefficient Immediately after the motion starts,
From this inequality, it may be observed that -
the curve is convex or concave from below, -
the total energy of the system is positive or negative, -
the virial coefficient of the system is or unity.
For small values of ( Proceeding as in Szebehely (1974a), we can show that the minimum
value of It is also true for unequal masses as well. Moreover, this is more general than Szebehely's (1974b) result viz 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 ( © European Southern Observatory (ESO) 1999 Online publication: May 21, 1999 |