Abstract
In a previous publication (1977) the author has constructed a family ℒ(α) of long-periodic orbits in the Trojan case of the restricted problems of three bodies. Here he constructs the domain
of the analytical solution of the problem of the motion, excluding the vicinity of thecritical divisor which vanishes at the exact commensurability of the natural frequencies ω1 and ω2. In terms of thecritical masses mj(α2), or the associatedcritical energies α 2j (m),
is the intersection of the intervals ofshallow resonance, of the form
. Inasmuch as the intervals |α2−α 2j |<εj ofdeep resonance aredisjoint, it follows that (1) the disjointed family ℒ(α) embraces the tadpole branch, 0⩽α2⩽1, lying in
: and (2) despite the clustering of α 2j (m) atj=∞, the family ℒ(α) includes, for α2=1, an asymptoticseparatrix that terminates the branch in the vicinity of the Lagrangian pointL 3.
In a similar manner, the family ℒ(α) can be extended to the horseshoe branch 1<α2⩽α 22 .
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Garfinkel, B. Theory of the Trojan asteroids, IV. Celestial Mechanics 30, 373–383 (1983). https://doi.org/10.1007/BF01375507
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DOI: https://doi.org/10.1007/BF01375507