Numerical simulations have also been carried out for the stability of P-type orbits in binary-planetary systems (Ziglin, 1975; Szebehely & McKenzie, 1981; Dvorak, 1984, 1986; Dvorak, Froeschle, & Froeschle, 1989; Kubala, Black & Szebehely, 1993; Holman & Wiegert, 1999; Broucke, 2001; Pilat-Lohinger, Funk & Dvorak, 2003; Musielak et al., 2005). Similar to S-type orbits, in order for a P-type planet to be stable, it has to be at a safe distance from the two stars so that it would be immune from their perturbative effects. That is, planets at large distances from the center of mass of a binary will have a better chance of being stable. This distance, however, cannot be too large because at very large distances, other astronomical effects, such as galactic perturbation, and perturbations due to passing stars, can render the orbit of a planet unstable.

To determine the critical value of the semimajor axis of a P-type planet in a stable orbit, preliminary attempts were made by Dvorak (1984), who numerically integrated the orbit of a circumbinary planet in a circular orbit around an eccentric binary system and showed that planets at distances 2-3 times the separation of the binary have stable orbits. Subsequent studies by Dvorak (1986), Dvorak, Froeschle, & Froeschle (1989), and Holman & Wiegert (1999) complemented Dvorak's results of 1984 and showed that the orbit of a P-type planet will be stable as long as the semimajor axis of the planet stays larger than the critical value given by (Fig. 9.10)

ac/ab = (1.60 ± 0.04) + (5.10 ± 0.05)eb + (4.12 ± 0.09)^

+ (-2.22 ± 0.11)e2 + (-4.27 ± 0.17)ebp + (-5.09 ± 0.11)^2 + (4.61 ± 0.36)e2^2 . (9.4)

Similar to Eq. (9.1), Eq. (9.4) represents a transitional region with a lower boundary below which the orbit of a P-type planet will be certainly unstable, and an upper

Fig. 9.10. Critical semimajor axis as a function of the binary eccentricity in a P-type system (Holman & Wiegert, 1999). The squares correspond to the result of stability simulations by Holman & Wiegert (1999) and the triangles represent those of Dvorak, Froeschle, & Froeschle (1989). The solid line corresponds to Eq. (9.4). As indicated by Holman & Wiegert (1999), the figure shows that at outer regions, the stability of the system fades away.

Fig. 9.10. Critical semimajor axis as a function of the binary eccentricity in a P-type system (Holman & Wiegert, 1999). The squares correspond to the result of stability simulations by Holman & Wiegert (1999) and the triangles represent those of Dvorak, Froeschle, & Froeschle (1989). The solid line corresponds to Eq. (9.4). As indicated by Holman & Wiegert (1999), the figure shows that at outer regions, the stability of the system fades away.

boundary beyond which the orbit of the planet will be stable. The mixed zone between these two boundaries represents a region where a planet, depending on its orbital parameters, and the orbital parameters and the mass-ratio of the binary, may or may not be stable. Recently, by applying the stability criteria of Eq. (9.4) to their observational results, Trilling et al. (2007) have confirmed the presence of stable dust bands, possibly resulting from collision of planetesimals, around close binary star systems.

A dynamically interesting feature of the stable region around the stars of a binary is the appearance of islands of instability. As shown by Holman & Wiegert (1999) islands of instability may develop beyond the inner boundary of the mixed zone, which correspond to the locations of (n : 1) mean-motion resonances. The appearance of these unstable regions have been reported by several authors under various circumstances (Henon & Guyot, 1970; Dvorak, 1984; Rabl & Dvorak, 1988; Dvorak, Froeschle, & Froeschle, 1989). Extensive numerical simulations would be necessary to determine whether the overlapping of these resonances would result in stable P-type binary-planetary orbits.

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