About one case of the nonrestricted three-body problem
E. Pittich
Astronomical Institute of the Slovak Academy of Sciences - Bratislava,
Slovak Republic
The present study is the investigation of particular case of the motion
of three points, in which masses of the points are comparable, and
the ratio of the semi-major axis of their orbits is the small parameter.
Differential equations of the motion are transformed by the Zeipel's
method. The Hamiltonian of the system are expended in terms of
the Legendre polynomial. Short-period terms are received to the fourth
order. For the analytical transformations were used the software computer
system Mathematica.
When the short-period terms were excluded from the Hamiltonian and taken
account only first three terms, the general solution of the dynamics
system was obtained in term of hyperelliptic integrals. The received
solution presents secular and long-period perturbations.
The triple system xi Ursa Maioris, whose components move along
short-period orbits with periods of 2 years (the inner orbits) and
60 years (outer orbit) was picked up as the interesting object
for applications. The results received from the analytical theory were
compared with those obtained by numerical simulations.