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

摘要

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. These systems have always a negative energy integral h and a large angular momentum c (in the axes of the center of masses), they are characterized by a product he2 smaller khan or equal to that of the corresponding circular Euter motion with the same three masses. They have a "close binary" and a "third body" that can neither approach nor disrupt the close binary (well defined limit distances can be given in terms of the three masses and the initial conditions). However, and this is a major difference with the circular restricted three-body problem, the third body can sometimes escape to infinity. 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. The third body always rotate into the positive direction about the direction of angular momentum, and many general properties of its orbit can be obtained: small inclination, upper and lower limits of its angular momentum, lower limits of its approach to the close binary, etc. But much less results are known for the relative orbit of the close binary (inner orbit), besides its roundedness. If the mutual inclination of the inner and outer orbits is small (or in the vicinity of 180°), they have generally only small and slow perturbations. But if that inclination is large, in the vicinity of 90°, the perturbations can become very large, especially for the eccentricity of the inner orbit and for the mutual inclination itself. Beside the usual types of orbits: bounded, with a parabolic or hyperbolic escape of the third body, etc. we must notice the presence of the two types of oscillating orbits. The first type is infinitely rare: the motion of the third body has then an infinite number of larger and larger loops, the reunion of which is unbounded, but it always come back to small distances. On the contrary the oscillating orbits of the second type fill in phase space a set of positive measure. These orbits are characterized by an infinite number of very close approaches of the two bodies of the close binary (their mutual distance has no positive lower bound and then their velocities are unbounded), even if strict collisions of point-masses remain infinitely rare. Of course real bodies are not point-masses and thus oscillating orbits of the second type lead to collisions. 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). The phenomenon of supernova requires an energy much larger than that of the usual collision of two stars, but the phenomenon of nova has an energy of similar order of magnitude. It is then likely that a proportion of novae appear in this indirect way, having, by far, a much larger probability than that of direct collisions of stars.