Study of the Sailboat stable region for binary systens.
Authors: Pinheiro Tiago , Sfair Rafael , Vieira Ernesto 
Published: September 2022

&lt;p&gt;The sailboat region was first identified by Giuliatti Winter, et al. (2010) exploring the Pluto-&lt;br /&gt;Charon binary system, they identifed this unexpected stable region of S-type orbits around the dwarf&lt;br /&gt;planet Pluto located at a = (0.5d, 0.7d) and e = (0.2, 0.9), where a and e are the initial values&lt;br /&gt;of semi-major axis and eccentricity of particles, respectively and d is the separation of the binary.&lt;br /&gt;The sailboat is associated with a family &amp;#8221;BD&amp;#8221; of periodic orbits derived from the planar, circular,&lt;br /&gt;restricted three-body problem. In this work, we analyzed through numerical simulations the structure&lt;br /&gt;and stability of sailboat in hypothetical systems with different values of mass ratio and for several&lt;br /&gt;orbital configurations.&lt;br /&gt;To constrain the orbital parameters for sailboat regions, we numerically simulated several elliptic&lt;br /&gt;three-body problem, exploring a large range of initial conditions. We adopt dimensionless systems&lt;br /&gt;and the configuration for each simulation include a test particle in S-type orbit around the primary&lt;br /&gt;body and gravitational disturbed by the secondary massive body. We set the central body as a point&lt;br /&gt;of mass and a secondary with a mass equivalent to the mass ratio of the binary system (&amp;#181;), with its&lt;br /&gt;radius (r s ) defined as 10% of their Hill radius.&lt;br /&gt;We created hypothetical systems with different mass ratio in the interval &amp;#181; = [0.01, 0.30] in steps&lt;br /&gt;&amp;#8710;&amp;#181; = 0.01. The test particles were randomly distributed with semimajor axis in a = [0.45, 0.7],&lt;br /&gt;considering 1 the distance between the two main bodies, the eccentricities varied from 0 to 0.99, and&lt;br /&gt;initially the argument of the pericentre and inclination was set as 0&amp;#186;. We numerically integrated using&lt;br /&gt;the REBOUND package and IAS15 integrator (Rein &amp; Spiegel 2014) for 10 4 orbital periods of the&lt;br /&gt;binary.&lt;br /&gt;We analyzed the behaviour of the sailboat according to the eccentricity e of the secondary body,&lt;br /&gt;looking for the maximum value for which the particles remain stable. A final set of simulations was&lt;br /&gt;performed for different values of inclinations and argument of pericenter in order to determine the&lt;br /&gt;extreme values for the stability.&lt;br /&gt;Our results show the sailboat is robust and it exists for &amp;#181; = [0.01,0.27] and for large intervals&lt;br /&gt;of the argument of pericentre and inclination. This region of stability reaches its maximum size&lt;br /&gt;with an argument of pericenter at 0 &amp;#9702; and 180 &amp;#9702; . The sailboat region also is present for values of&lt;br /&gt;inclination &gt; 60&amp;#186; and existing even retrogrades orbits in the systems with &amp;#181; &gt; 0.08.&lt;br /&gt;The numerical results also showed that little changes in the eccentricity of the secondary body is&lt;br /&gt;sufficient to vanish the sailboat region, for binaries system with &amp;#181; &gt; 0.12, the sailboat exists just for&lt;br /&gt;values of e &lt; 0.05.&lt;/p&gt;