Study of the Sailboat stable region for binary systens.

The sailboat region was first identified by Giuliatti Winter, et al. (2010) exploring the Pluto-
Charon binary system, they identifed this unexpected stable region of S-type orbits around the dwarf
planet Pluto located at a = (0.5d, 0.7d) and e = (0.2, 0.9), where a and e are the initial values
of semi-major axis and eccentricity of particles, respectively and d is the separation of the binary.
The sailboat is associated with a family ”BD” of periodic orbits derived from the planar, circular,
restricted three-body problem. In this work, we analyzed through numerical simulations the structure
and stability of sailboat in hypothetical systems with different values of mass ratio and for several
orbital configurations.
To constrain the orbital parameters for sailboat regions, we numerically simulated several elliptic
three-body problem, exploring a large range of initial conditions. We adopt dimensionless systems
and the configuration for each simulation include a test particle in S-type orbit around the primary
body and gravitational disturbed by the secondary massive body. We set the central body as a point
of mass and a secondary with a mass equivalent to the mass ratio of the binary system (µ), with its
radius (r s ) defined as 10% of their Hill radius.
We created hypothetical systems with different mass ratio in the interval µ = [0.01, 0.30] in steps
∆µ = 0.01. The test particles were randomly distributed with semimajor axis in a = [0.45, 0.7],
considering 1 the distance between the two main bodies, the eccentricities varied from 0 to 0.99, and
initially the argument of the pericentre and inclination was set as 0º. We numerically integrated using
the REBOUND package and IAS15 integrator (Rein & Spiegel 2014) for 10 4 orbital periods of the
binary.
We analyzed the behaviour of the sailboat according to the eccentricity e of the secondary body,
looking for the maximum value for which the particles remain stable. A final set of simulations was
performed for different values of inclinations and argument of pericenter in order to determine the
extreme values for the stability.
Our results show the sailboat is robust and it exists for µ = [0.01,0.27] and for large intervals
of the argument of pericentre and inclination. This region of stability reaches its maximum size
with an argument of pericenter at 0 ◦ and 180 ◦ . The sailboat region also is present for values of
inclination > 60º and existing even retrogrades orbits in the systems with µ > 0.08.
The numerical results also showed that little changes in the eccentricity of the secondary body is
sufficient to vanish the sailboat region, for binaries system with µ > 0.12, the sailboat exists just for
values of e < 0.05.