Hill type stability

GENERAL PROPERTIES OF THREE-BODY SYSTEMS WITH HILL-TYPE STABILITY... 

Abstract Three-body systems with Hill-type stability are the generalization to the general three-body problem of the Hill-stable orbits of the circular restricted three-body problem. 

A large majority of known triple stellar systems have the Hill-type stability, and so is the Sun-Jupiter-Saturn system (99.99% of the mass of the Solar System). The close binary is then the Sun and Jupiter, while Saturn is the third body. 

Let us now have a physical point of view. In our galaxy a majority of stars are binary stars. If then a weak binary star meets a strong binary star, an ordinary motion of exchange type can easily disrupt the weak binary and lead to the formation of a triple system with the strong binary. That new-born triple system has generally a Hill-type stability and if its motion is of the second oscillating type (which usually requires a large inclination) it will lead to a collision of the two stars of the binary... The probability of this phenomenon is of the order of the ratio of the inner period (that of the close binary) to the outer period (that of the third body). 

Keywords Celestial Mechanics. 3 body-problem. Hill-type stability. 

General properties of three-body systems with Hill-type stability... 

If (p/a)lb

phase space corresponding to the given integrals of motion is disconnected into two parts a part about mA (Hill type stability, with the two largest masses mg and mc as the close binary and m as the third outer body) and a second part about mg, LA and me- Notice that, even if p/a = p/a(Lg), no motion can cross Lg. 

That limit ratio is always smaller than unity, and thus we have ever r < R in motions with Hill type stability. 

A) For a motion with the Hill type stability we must add the second condition (22) ... 

Two lengths r and R are given and satisfy (32) in a three-body problem with Hill type stability. Are there some suitable orientations that satisfy (31) or (33) The optimal orientations are with collinear r and R, in the same direction, and both normal to c. 

Let us consider a three-body system with Hill type stability. The corresponding R, r point is of course in the zone of possible motion of a suitable figure as Figure 7 (conditions (32) and (34)) and the corresponding radius vectors R and r and velocity vectors V and v satisfy the equations (27) and (28) of the two integrals of motion. However notice that, in order to satisfy the condition (34) it is sufficient to find two velocity vectors V and v such that ... 

Let us also add the two following propositions A) This test is valid even for three-body systems without the Hill type stability, even if the energy integral h is positive. B) A condition similar to (54) allows to extend the test to t/ = oo and to give a test of escape. R is sufficient that ... 

This phenomenon is very general when the inclinations U and j are far from 0° and 180°, but the plane motions with Hill type stability undergo only small and slow perturbations. 

In spite of a greater simplicity than the general three-body problem, and especially in spite of the absence of motions of exchange type, the three-body systems with Hill type stability have a wide variety of solutions and, when the mutual inclination is large, they undergo large perturbations that can sometimes lead to the collision of the two bodies of the binary. 

The complexity of all these results give a small taste of the far larger complexity of the general three-body problem with possible escape of any of the three bodies. It would be especially of great interest to look for motions of exchange type and to determine how close they can approach to the Hill type stability. 

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