As mentioned above, instability occurs when the perturbative effects cause the semimajor axis and orbital eccentricity of a planet change in such a way that either the object leaves the gravitational field of the system, or it collides with another body. For a planet in an S-type orbit, the gravitational force of the secondary star is the source of these perturbations. That implies, a planet at a large distance from the secondary, i.e., in an orbit closer to its host star, may receive less perturbation from the binary companion and may be able to sustain its dynamical state for a longer time (Harrington, 1977). Since the perturbative effect of the stellar companion varies with its mass, and the eccentricity and semimajor axis of the binary

3Symplectic integrators, as they were originally developed by Wisdom & Holman (1991), are not suitable for numerically integrating the orbits of small bodies in the gravitational fields of two massive objects. These integrators have been designed to integrate the orbits of planetary or smaller objects when they revolve around only one massive central body. Recently Chambers et al. (2002) have developed a version of a symplectic integrator that is capable of integrating the motion of a small object in the gravitational fields of two stellar bodies.

hinary e M

Fig. 9.5. Graphs of the critical semimajor axis (ac) of an S-type binary-planetary system, in units of the binary semimajor axis (Holman & Wiegert, 1999). The graph on the left shows &c as a function of the binary eccentricity for an equal-mass binary. The graph on the right corresponds to the variations of the critical semimajor axis of a binary with an eccentricity of 0.5 in term of the binary's mass-ratio. The solid and dashed line on the left panel depict the empirical formulae as reported by Holman & Wiegert (1999) and Rabl & Dvorak (1988), respectively.

hinary e M

Fig. 9.5. Graphs of the critical semimajor axis (ac) of an S-type binary-planetary system, in units of the binary semimajor axis (Holman & Wiegert, 1999). The graph on the left shows &c as a function of the binary eccentricity for an equal-mass binary. The graph on the right corresponds to the variations of the critical semimajor axis of a binary with an eccentricity of 0.5 in term of the binary's mass-ratio. The solid and dashed line on the left panel depict the empirical formulae as reported by Holman & Wiegert (1999) and Rabl & Dvorak (1988), respectively.

(which together determine the closest approach of the secondary to the planet), it is possible to estimate an upper limit for the planet's distance to the star beyond which the orbit of the planet would be unstable. As shown by Rabl & Dvorak (1988) and Holman & Wiegert (1999), the maximum value that the semimajor axis of a planet in an S-type orbit can attain and still maintain its orbital stability is a function of the mass-ratio and orbital elements of the binary, and is given by (Rabl & Dvorak, 1988; Holman & Wiegert, 1999)

ac/ab = (0.464 ± 0.006) + (-0.380 ± 0.010) p + (-0.631 ± 0.034)eb

+(0.586 ± 0.061)peb + (0.150 ± 0.041)eb + (-0.198 ± 0.047)peb . (9.1)

In this equation, ac is critical semimajor axis, p = M\/(M\ + Mb), ab and eb are the semimajor axis and eccentricity of the binary, and Mi and Mb are the masses of the primary and secondary stars, respectively. The ± signs in Eq. (9.1) define a lower and an upper value for the critical semimajor axis ac, and set a transitional region that consists of a mix of stable and unstable systems. Such a dynamically gray area, in which the state of a system changes from stability to instability, is known to exist in multi-body environments, and is a characteristic of any dynamical system.

Equation (9.1) is an empirical formula that has been obtained by numerically integrating the orbit of a test particle (i.e., a massless object) at different distances from the primary of a binary star (Rabl & Dvorak, 1988; Holman & Wiegert, 1999). Figure 9.5 shows this in more detail. Similar studies have been done by Moriwaki &

30 20 10 0

30 20 10 0

Ecc.Bin.=0.25 Inclination=0

Ecc.Bin.=0.45 Inclination=0

2.74 5.48 8.22 Million Years

10.96

2.74 5.48 8.22 Million Years

10.96

2.74 5.48 8.22 Million Years

10.96

Fig. 9.6. Graphs of the semimajor axes (left) and eccentricities (right) of the giant planet (black) and binary (green) of 7 Cephei for different values of the eccentricities of the binary (Haghighipour, 2003). The mass-ratio of the binary is 0.2.

Nakagawa (2004), and Fatuzzo et al. (2006) who obtained critical semimajor axes slightly larger than given by Eq. (9.1).

Since the mass of a Jovian-type planet is approximately three orders of magnitude smaller than the mass of a star, such a test particle approximation yields results that are not only applicable to the stability of giant planets, but can also be used in identifying regions where smaller bodies, such as terrestrial-class objects (Quintana et al., 2002; Quintana & Lissauer, 2006; Quintana et al., 2007) and dust particles (Trilling et al., 2007), can have long term stable orbits4. In a recent article, Trilling et al. (2007) utilized Eq. (9.1) and its stability criteria to explain the dynamics of debris disks, and the possibility of the formation and existence of plan-etesimals in and around 22 binary star systems. By detecting an infrared excess of dust particles, these authors confirmed the presence of stable dust bands, possibly resulting from collision of planetesimals, in S-type orbits in several wide binaries.

The stability of S-type systems has been studied by many authors (Benest, 1988, 1989, 1993, 1996; Wiegert & Holman, 1997; Pilat-Lohinger & Dvorak, 2002; Dvorak et al., 2003, 2004; Pilat-Lohinger et al., 2004; Musielak et al., 2005). In a recent article, Haghighipour (2006) extended such studies to the dynamical stability of the Jupiter-like planet of the 7 Cephei planetary system. By numerically integrating the orbit of this object for different values of ab, eb and ip (the orbital inclination of the giant planet relative to the plane of the binary), Haghighipour (2006) has shown that the orbit of this planet is stable for the values of the binary eccentricity within the range 0.2 < eb < 0.45. Figure 9.6 shows the results of such integrations for a coplanar system with j = 0.2 and for different values of the binary eccentricity. The initial value of the semimajor axis of the binary was chosen to be

4In applying Eq. (9.1) to the stability of dust particles, one has to note that this equation does not take into account the effects of non-gravitational forces such as gas-drag or radiation pressure. The motion of a dust particle can be strongly altered by the effects of these forces.

Inclination^ degrees |
Inclination=IO degrees |
Inclination^ degrees |

0-2 ammmmmmmmmmmm ■■■■■■■■■ — - ---- -

D 2.74 5.48 822 10.960 2.74 5.48 822 10.960 2.7+ 5.48 8.22 10.96 Million Years Million Years Million Years

0-2 ammmmmmmmmmmm ■■■■■■■■■ — - ---- -

D 2.74 5.48 822 10.960 2.74 5.48 822 10.960 2.7+ 5.48 8.22 10.96 Million Years Million Years Million Years

Fig. 9.7. Graphs of the semimajor axes (top) and eccentricities (bottom) of the giant planet (black) and binary (red) of 7 Cephei. The initial eccentricity of the binary at the beginning of numerical integration and the value of its mass-ratio were equal to 0.20 (Haghighipour, 2006).

21.5 AU. Integrations also indicated that the binary-planetary system of 7 Cephei becomes unstable in less than a few thousand years when the initial value of the binary eccentricity exceeds 0.5.

Interesting results were obtained when the 7 Cephei system was integrated for different values of ip. The results indicated that for the above-mentioned range of orbital eccentricity, the planet maintains its orbit for all values of inclination less than 40°. Figure 9.7 shows the semimajor axes and orbital eccentricities of the system for eb = 0.2 and for ip=5°, 10°, and 20°. For orbital inclinations larger than 40°, the system becomes unstable in a few thousand years.

An interesting dynamical phenomenon that may occur in an S-type binary, and has also been observed in the numerical simulations of a few of these systems, is the Kozai resonance (Haghighipour, 2003, 2005; Verrier & Evans, 2006; Takeda & Rasio, 2006; Malmberg, Davies & Chambers, 2007). As demonstrated by Kozai (1962), in a three-body system with two massive objects and a small body (e.g., a binary-planetary system), the exchange of angular momentum between the planet and the secondary star, can cause the orbital eccentricity of the planet to reach high values at large inclinations. Averaging the equations of motion of the system over mean anomalies, one can show that in this case, the averaged system is integrable when the ratio of distances are large (the Hill's approximation, Kozai, 1962). The Lagrange equations of motion in this case, indicate that, to the first order of planet's eccentricity, the longitude of the periastron of this object, up, librates around a fixed value. Figure 9.8 shows this for the giant planet of 7 Cephei. As shown here, up librates around 90° (Haghighipour, 2003, 2005).

In a Kozai resonance, the longitude of periastron and the orbital eccentricity of the small body (ep) are related to its orbital inclination as (Innanen et al., 1997)

Fig. 9.8. Graphs of the semimajor axis and eccentricity of the giant planet (black) and binary (red) of 7 Cephei (top) and its longitude of periastron and reduced Delaunay momentum (bottom) in a Kozai resonance (Haghighipour, 2003, 2005). As expected, the longitude of the periastron of the giant planet oscillates around 90° and its reduced De-launay momentum is constant.

Fig. 9.8. Graphs of the semimajor axis and eccentricity of the giant planet (black) and binary (red) of 7 Cephei (top) and its longitude of periastron and reduced Delaunay momentum (bottom) in a Kozai resonance (Haghighipour, 2003, 2005). As expected, the longitude of the periastron of the giant planet oscillates around 90° and its reduced De-launay momentum is constant.

From Eq. (9.3), one can show that the Kozai resonance may occur if the orbital inclination of the small body is larger than 39.23°. For instance, as shown by Haghighipour (2003, 2005), in the system of 7 Cephei, Kozai resonance occurs at ip = 60°. For the minimum value of ip, the maximum value of the planet's orbital eccentricity, as given by Eq. (9.3), is equal to 0.764. Figure 9.8 also shows that ep stays below this limiting value at all times.

As shown by Kozai (1962) and Innanen et al. (1997), in a Kozai resonance, the disturbing function of the system, averaged over the mean anomalies, is independent of the longitudes of ascending nodes of the small object (the planet) and the perturbing body (the stellar companion). As a result, the quantity a(1 — e2) cos i (shown as the "Reduced Delaunay Momentum" in Fig. 9.8) becomes a constant of motion. Since the eccentricity and inclination of the planet vary with time, the fact that the quantity above is a constant of motion implies that the time-variations of these two quantities have similar periods and, at the same time, they vary in such a way that when ip reaches its maximum, ep reaches its minimum and vice versa. Figure 9.9 shows this clearly.

Fig. 9.9. Graphs of the eccentricity and inclination of the giant planet of 7 Cephei in a Kozai Resonance (Haghighipour, 2003, 2005). As expected, these quantities have similar periodicity and are 180° out of phase.

(10000 Years)

Fig. 9.9. Graphs of the eccentricity and inclination of the giant planet of 7 Cephei in a Kozai Resonance (Haghighipour, 2003, 2005). As expected, these quantities have similar periodicity and are 180° out of phase.

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