Distance Au

Fig. 9.23. Habitable zone (Kasting, Whitmire, & Reynolds, 1993).

Fig. 9.24. Graph of the lifetime of an Earth-size object in a circular orbit around the primary of y Cephei. The habitable zone of the primary has been indicated by HZ. No planet was placed in the region between the apastron and periastron distances of the giant planet of the system. As shown here, only Earth-size planets close to the primary star maintain their orbits for long times (Haghighipour, 2006).

Fig. 9.24. Graph of the lifetime of an Earth-size object in a circular orbit around the primary of y Cephei. The habitable zone of the primary has been indicated by HZ. No planet was placed in the region between the apastron and periastron distances of the giant planet of the system. As shown here, only Earth-size planets close to the primary star maintain their orbits for long times (Haghighipour, 2006).

Equation (9.6) implies that a habitable zone can be defined as a region around a star where an Earth-like planet can receive the same amount of radiation as Earth receives from the Sun, so that it can develop and maintain similar habitable conditions as those on Earth.

As mentioned above, the orbit of a potential habitable planet in the habitable zone of a star has to be stable over long durations of time. As shown in Sect. 9.2, the stability of the orbit of a planet in a binary system is strongly affected by the orbital motion of the binary companion. In binary systems where the primary hosts other planetary bodies (e.g., giant planets), the dynamics of a habitable planet will also be affected by the gravitational perturbations of these objects. It is therefore important to determine under what conditions a terrestrial-class object will have a long-term stable orbit in the habitable zone of a binary system, prior to constructing a theory for the formation of Earth-like planets in such environments.

Since a terrestrial-class planet is approximately two orders of magnitude less massive than a Jovian-type object, it will not have a significant effect on the motion of the stars and the giant planets of a binary system. Therefore, as explained in Sects. 9.2.1 and 9.2.2, if a binary system does not contain a Jupiter-like planet, any dynamical criterion that is obtained for the stability or instability of a general planetary body, can also be applied to the dynamics of a terrestrial planet. Equation (9.1) and the stability conditions presented by Fig. 9.5 can be used to determine the long-term stability of an Earth-like object in a binary system.

If a binary contains giant planets, however, the situation is different. The gravitational perturbations of the latter objects will have significant effects on the motion and dynamics of terrestrial planets in the system. As shown by Haghighipour (2006), an Earth-size object, in a region between the giant planet and the primary of Y Cephei binary system, can maintain its stability only in orbits close to the primary star and outside the influence zone5 of the giant body. Integrating the equations of motion of a full four-body system, this author has shown that an Earth-like planet will not be able to sustain a stable orbit in the habitable zone of y Cephei's primary star (Fig. 9.24). However, it is possible for such an object to have a stable orbit when 0.3 < aT < 0.8 AU, 0° < iT = ip < 10°, and eb < 0.4. Here aT represents the semimajor axis of the terrestrial planet and iT is its orbital inclination with respect to the plane of the binary.

As mentioned above, the instability of an Earth-like planet in the habitable zone of Y Cephei can be attributed to the interaction between this object and the giant planet of the system. When the Earth-like planet is outside the giant planet's influence zone (e.g., at closer distances to the primary star) it can maintain its orbit for several hundred million years. Figure 9.24 suggests that, in order for a binary-planetary system to be habitable, its habitable zone has to be outside the influence region of its giant planet. In an S-type binary-planetary system, this implies a primary with a close-in habitable region. In a recent article, Haghighipour

5The influence zone of a planetary object with a mass mp around a star with a mass M is defined as the region between 3RH — ap(1 — ep) and 3RH + ap(1 + ep), where ap is the semimajor axis of the planet, and Rh = ap(mp/3M)1//3 is its Hill radius

Fig. 9.25. Formation of Earth-like planets in a binary-planetary system. The top panel shows simulations in a binary with a separation of 30 AU, eccentricity of 0.2, and stellar components of 1 solar-mass. As shown here, an Earth-like planet (1.17 Earth-masses) with a water to mass ratio of 0.00164, is formed in the habitable zone of the primary at 1.16 AU, with an eccentricity of 0.02. The bottom panel shows the formation of an Earth-like object in a binary with a solar-mass primary and a 1.5 solar-masses secondary. The separation of the binary in this case is 30 AU, the mass of the Earth-like planet is 0.95 Earth-masses and its water to mass ratio is 0.00226. The semimajor axis of this planet and its orbital eccentricity are equal to 0.99 AU and 0.07, respectively. For the sake of comparison, the Sun's habitable zone is approximately at 0.95-1.15 AU, Earth's orbital eccentricity is 0.017, and Earth's water to mass ratio is ~ 0.001 (Haghighipour & Raymond, 2007).

Fig. 9.25. Formation of Earth-like planets in a binary-planetary system. The top panel shows simulations in a binary with a separation of 30 AU, eccentricity of 0.2, and stellar components of 1 solar-mass. As shown here, an Earth-like planet (1.17 Earth-masses) with a water to mass ratio of 0.00164, is formed in the habitable zone of the primary at 1.16 AU, with an eccentricity of 0.02. The bottom panel shows the formation of an Earth-like object in a binary with a solar-mass primary and a 1.5 solar-masses secondary. The separation of the binary in this case is 30 AU, the mass of the Earth-like planet is 0.95 Earth-masses and its water to mass ratio is 0.00226. The semimajor axis of this planet and its orbital eccentricity are equal to 0.99 AU and 0.07, respectively. For the sake of comparison, the Sun's habitable zone is approximately at 0.95-1.15 AU, Earth's orbital eccentricity is 0.017, and Earth's water to mass ratio is ~ 0.001 (Haghighipour & Raymond, 2007).

Fig. 9.26. Formation of Earth-like planets in different binary-planetary systems. As shown here, for a given binary mass-ratio, the delivery of water to terrestrial regions becomes less efficient as the periastron distance of the binary becomes smaller (Haghighipour & Raymond, 2007).

& Raymond (2007) have studied the habitability of such a system. By considering a binary with a Sun-like primary star and a Jupiter-sized planet in a circular orbit at 5 AU, and by adopting the model of Morbidelli et al. (2000), which is based on the assumption that water-carrying objects, in the Sun's asteroid belt, were the primary source of the delivery of water to Earth, these authors integrated the orbits of a few hundred protoplanetary (Moon- to Mars-sized) objects, and showed that it is indeed possible to form Earth-sized planets, with substantial amounts of water, in the habitable zone of the primary star (Fig. 9.25). As shown by these authors, the mass and orbital parameters of the secondary star play important roles in the radial mixing of protoplanetary objects and the delivery of water to the habitable zone of the primary star. The giant planet of the system also plays the important role of transferring angular momentum from the secondary star to the disk of protoplanets, and enhancing the radial mixing of these objects. As shown in Fig. 9.26, water delivery is less efficient in binaries with smaller perihelia since

▼ Unstable

y

• Stable

y

• ■

• ■

y' 0.03

0.02

Binary Semimajor Axis (AU)

20 30 40

Binary Semimajor Axis (AU)

Fig. 9.27. The (eb,ab) parameter-space of an equal-mass binary-planetary system. Circles correspond to binaries with initial parameters chosen from Fig. 9.25, in which habitable planets were formed. Triangles represent systems in which the giant planet became unstable. The numbers associated with each circle represent the mean eccentricity of the giant planet of the system at the end of the simulation. As shown here, moderately close binaries with lower eccentricities (larger periastra) are more suitable places for the formation of habitable planets (Haghighipour & Raymond, 2007).

in such systems, the close approach of the binary companion to the giant planet increases its eccentricity, which in turn results in stronger interaction between this object and the disk of protoplanets, causing them to become unstable in a very short time. The results of the simulations by Haghighipour & Raymond (2007) indicate that binary-planetary systems with giant planets at 5-10 AU, and binary perihelion distances of approximately 20 AU to 25 AU, will be more efficient in forming and hosting habitable planets (Fig. 9.27).

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