5. Escape velocity in terms of semimajor axis
The asymptotic hyperbolic escape velocity relative to the centre of mass of the general gravitational three bodies and its asymptotic value of the semimajor axis of the binary depends on the perturbing velocities (i = 1, 2, 3). For these small values of and any values of their directions (i = 1, 2, 3), we have
To show it, first attention is directed to the double limit process involved in the above result. After the binary is formed, the distance r between the escaper and the centre of mass of the binary, the velocity of the escaper, and the semimajor axis of the binary `a' approaches asymptotically to the value In this limit process the original three-body problem approaches its partition into two two-body problems. The escaper and the centre of mass of the binary form a hyperbolic two-body problem and the members of the binary form another elliptic two-body problem.
The first limit process refers to the behaviour of the general gravitational three-body problem as the perturbing velocities approach zero and we consider the general form of the total energy
The second limit process refers after the binary is formed. The total energy, may be written as
where is the escape energy, is the energy stored in the binary, and is the correction due to three-body effects. Equations for and may be written from two-body consideration as follows:
where and form the binary and M is the total mass of the participating bodies. In this limit process is fixed,
Equating the above two forms of the total energy we have
It is also true for unequal masses as well. Moreover this is more general than Szebehely's (1974a,b) result viz., which can be deduced from our results.
© European Southern Observatory (ESO) 1999
Online publication: May 21, 1999