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

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5. Escape velocity in terms of semimajor axis

The asymptotic hyperbolic escape velocity [FORMULA] relative to the centre of mass of the general gravitational three bodies and its asymptotic value of the semimajor axis of the binary [FORMULA] depends on the perturbing velocities [FORMULA] (i = 1, 2, 3). For these small values of [FORMULA] and any values of their directions [FORMULA] (i = 1, 2, 3), we have [FORMULA]

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, [FORMULA] the velocity of the escaper, [FORMULA] and the semimajor axis of the binary `a' approaches asymptotically to the value [FORMULA] 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

[EQUATION]

The second limit process refers after the binary is formed. The total energy, [FORMULA] may be written as

[EQUATION]

where [FORMULA] is the escape energy, [FORMULA] is the energy stored in the binary, and [FORMULA] is the correction due to three-body effects. Equations for [FORMULA] and [FORMULA] may be written from two-body consideration as follows:

[EQUATION]

where [FORMULA] and [FORMULA] form the binary and M is the total mass of the participating bodies. In this limit process [FORMULA] is fixed,

[EQUATION]

or [FORMULA]

Equating the above two forms of the total energy we have

[EQUATION]

or [FORMULA]

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

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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