Stability of orbits near large mass ratio binary systems

Abstract

With improved observational capabilities and techniques, an increasing number of exoplanets have been discovered to orbit in the vicinity of binary star systems. In this investigation, periodic motions near a large mass ratio binary are explored within the context of the circular restricted three-body problem. Specifically, stability analysis is used to explore the effect of the mass ratio on the structure of families of periodic orbits. Such analysis is useful in a variety of applications, including the determination of potentially stable exoplanet motions near a binary star.

Appendix: Initial conditions for representative orbits in each family

For each planar symmetric family examined in this investigation, truncated initial conditions are provided for a representative orbit at a single value of the mass ratio, $\mu =0.30$. Each initial condition, $\bar{x}_{IC}$, takes the form of a six dimensional state located at one of the two perpendicular intersections of the orbit with the $x$-axis, such that $\bar{x}_{IC} = [x_{0}, 0, 0, 0, \dot{y}_{0}, 0]$. Integrating $\bar{x}_{IC}$ for a time interval equal to the orbital period, $T$, produces a periodic orbit. Accordingly, each of the nonzero state elements, $x_{0}$ and $\dot{y}_{0}$, as well as the period are provided in Table 1 in nondimensional units. Since the results of a numerical integration in a chaotic system depend upon the specific implementation (i.e., choice of integration scheme and tolerance, as well as numerical accuracy) these initial conditions are provided in truncated form and require differential corrections to recover an orbit that is periodic, to within a specified tolerance.

Table 1 Initial conditions for a representative orbit along each planar, symmetric family at $\mu =0.30$