## 5. Escape velocity in terms of semimajor axisThe 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
( To show it, first attention is directed to the double limit process
involved in the above result. After the binary is formed, the distance
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 or Equating the above two forms of the total energy we have or 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 |