## Dynamical Evolution and Stability

In general, one can consider orbital stability synonymous with the capability of an object in maintaining its orbital parameters (i.e., semimajor axis, eccentricity, and inclination) at all times. In other words, an object is stable if small variations in its orbital parameters do not progress exponentially, but instead vary sinusoidally. Instability occurs when the perturbative forces create drastic changes in the time variations of these parameters and result in either the ejection of the object from the system (i.e., leaving the system's gravitational field), or its collision with other bodies.

The concept of stability, as explained above, although simple, has been defined differently by different authors. A review by Szebehely in 1984 lists 50 different definitions for the stability of an object in a multi-body system (Szebehely, 1984). For instance, Harrington (1977) considered the orbit of an object stable if, while numerically integrating the object's orbit, its semimajor axis and orbital eccentricity would not undergo secular changes. Szebehely (1980), and Szebehely & McKenzie (1981), on the other hand, considered integrals of motion and curves of zero velocity to determine the stability of a planet in and around binary stars. In this chapter, the values of the orbital eccentricity of an object and its semimajor axis are used to evaluate its stability. An object is considered stable if, for the duration of the integration of its orbit, the value of its orbital eccentricity stays below unity, it does not collide with other bodies, and does not leave the gravitational field of its host star.

The study of the stability of a planetary orbit in dual stars requires a detailed analysis of the dynamical evolution of a three-body system. Such an analysis itself is dependent upon the type of the planetary orbit. In general, a planetary-class object may have three types of orbit in and around a binary star system. Szebehely (1980) and Dvorak (1982) have divided these orbits into three different categories. As indicated by Szebehely (1980), a planet may be in an inner orbit, where it revolves around the primary star, or it may be in a satellite orbit, where it revolves around the secondary star. A planet may also be revolving the entire binary system in which case its orbit is called an outer orbit. As classified by Dvorak (1982), on the other hand, a study of the stability of resonant periodic orbits in a restricted, circular, three-body system indicates that a planet may have an S-type orbit, where it revolves around only one of the stars of the binary, or may be in a P-type orbit, where it revolves the entire binary system (Fig. 9.4). A planet may also be in an L-type orbit where it librates in a stable orbit around the L4 or L5 Lagrangian points.

Fig. 9.4. S-type and P-type binary-planetary systems. A and B represent the stars of the binary, and P depicts the planet.

S-type

The rest of this section is devoted to a review of the studies of the stability of planets in S-type and P-type orbits. Given that in a binary star system, a planet is subject to the gravitational attraction of two massive bodies (i.e., the stars), it would be important to understand how the process of the formation of planetary objects, and the orbital dynamics of small bodies would be affected by the orbital characteristics of the binary's stellar components. In general, except for some special cases for which analytical solutions may exist, such studies require numerical integrations of the orbits of all the bodies in the system. In the past, prior to the invention of symplectic integrators (Wisdom & Holman, 1991), which enabled dynamicists to extend the studies of the stability of planetary systems to several hundred million years, and before the development of fast computers, the majority of such studies were either limited to those special cases, or were carried out numerically for only a few orbital periods of a binary. Examples of such studies can be found in articles by Graziani & Black (1981), Black (1982), and Pendleton & Black (1983), in which the authors studied the orbital stability of a planet in and around a binary star. By numerically integrating the equations of the motion of the planet, these authors showed that, when the stars of the binary have equal masses, the orbital stability of the planet is independent of its orbital inclination (also see Harrington, 1977). Their integrations also indicated that, when the mass of one of the stellar components is comparable to the mass of Jupiter, planetary orbits with inclinations higher than 50° tend to become unstable.

The invention of symplectic integrators, in particular routines that have been designed specifically for the purpose of integrating orbits of small bodies in dualstar systems3 (Chambers et al., 2002), have now enabled dynamicists to extend studies of planet formation and stability in dual-star environments to much larger timescales. In the following, the results of such studies are discussed in more detail.