ON THE POSSIBILITY OF ADDITIONAL PLANETS IN THE γ CEPHEI BINARY-PLANETARY SYSTEM Joseph Castro Mentor: Nader Haghighipour ABSTRACT Results of the simulations of the dynamical stability of additional hypothetical planets in the γ Cephei binary-planetary system are presented. Using the reported orbital parameters of the existing bodies (Hatzes et al. 2003), simulations were carried out for different values of the mass, eccentricity, and inclination of an additional Jupiter-like plantet to test the stability of this object at a distance of 3 to 5 AU from the primary star. Simulations were also carried out for fictitious Saturn- and Neptune-like planets. Results indicate that the system appears to be dynamically full and no stable orbits of any additional planets were found. 1. INTRODUCTION In 1992, the first discovery of an extra-solar planet was made around the pulsar PSR 1257 (Wolszczan and Frail 1992). It was not until three years later on October 6, 1995, that an extra-solar planet was discovered orbiting a Sun-like star (Mayor and Queloz 1995). Since then, astronomers have been finding extra-solar planets at a maddening rate, and the tally is now up to over 190. Among these planets, approximately 25% are members of binary or multistar systems. These systems, for the most part, have very large stellar separations, generally between 250 and 6500 AU; distances at which the gravitational effect of the farther companion is insignificant. When the separation is less than 100 AU, however, the farther companion can have profound effects on the dynamics of the system. Among the three known systems of this type, this paper focuses on the binary-planetary system of γ Cephei. Gamma Cephei is a spectroscopic binary located approximately 50 light-years from Earth. This system has a 1.59 solar-mass K1 IV subgiant as its primary star, and a 0.34-0.92 solar-mass red giant as its secondary. The semimajor axis and eccentricity of this binary are 18.5 ± 1.1 AU and 0.361 ± 0.023, respectively (Hatzes et al. 2003). A Jupiter- like planet with at least 1.7 Jupiter-masses, orbits the system’s primary star in an orbit with a semimajor axis of 2.13 ± 0.05 AU and an eccentricity of 0.12 ± 0.05 (Hatzes et al. 2003). Table 1 shows the six orbital parameters of the γ Cephei system as reported by Hatzes et al. (2003). Gamma Cephei is an interesting case in that, it is not only the first of three moderately close planet-hosting binary/multistar systems, it also could have been the very first extra-solar planetary system ever discovered. In 1988, a Canadian team led by Bruce Campbell and Gordon Walker, suspected the existence of a third body in the binary system of γ Cephei while studying the radial velocity variations of this system. In 1992, however, the possibility that the RV variations observed by the authors were due to the long-period RV variability of most K giants stars (under which γ Cephei is classified) was brought about, and their claim of the presence of an extra-solar planet was retracted. Finally, ten years later, using new radial velocity measurements made by the McDonald Obseravtory, the existence of the planet was confirmed Studies have already been carried out on the stability of the planet of γ Cephei system. In 2003, a study by Dvorak et al. (2003) confirmed the dynamical stability of this planet for a period of approximately 108 years. The stability of this object was also confirmed separately by Haghighipour (2006). Using a range of values for the eccentricity and semimajor axis of the binary as well as various inclinations of the Jupiter-like planet, Haghighipour (2006) found that the orbit of this planet is stable when the value of the binary eccentricity is smaller than 0.5, and the inclination of the planet is between 0◦ and 40◦. Studies have also been carried out to determine whether additional planets and asteroids can exist within the γ Cephei system. Verrier and Evans (2006) performed numerical simulations to study the stability of test particles in this system and were able to identify possible locations where additional planets and asteroids could be stable. These authors also studied the stability of objects around each star (interior and exterior to the planet) as well as the circumbinary case. Around the primary star, these authors identified three zones of stability for at least 1-Myr, from 0.5 to 0.75 AU, 1.05 to 1.15 AU, and also 1.2 to 1.3 AU. Around the secondary, they identified a region of stability at approximately 0.5 to 1.5 AU from this star. Finally, in a region around both stars, they determined a zone of stability TABLE 1 Orbital Parameters of the γ Cephei System (Hatzes et al. 2003) Parameter Planet Binary a (AU) 2.13 ± 0.05 18.5 ± 1.1 e 0.12 ± 0.05 0.361 ± 0.023 i (deg) 0 0 Ω (deg) 0 0 ω (deg) 49.6 ± 25.6 158.76 ± 1.2 M (deg) 104 353 2 TABLE 2 Parameter Known Planet Extra Planet Binary Mass 1.7 MJ 0.25-2.5 MJ 0.4 M⊙ a (AU) 2.13 3-5 18.5 e 0.12 0.0-0.2 0.361 i (deg) 0 0-10 0 Ω (deg) 0 0 0 ω (deg) 49.6 0 158.76 M (deg) 104 0 353 extending out from near 65 AU. Instead of using test particles, Haghighipour (2006) simulated the dynamics of an actual Earth-like planet. Con- sidering a range of 0.3 to 4.0 AU for the semimajor axis of this object, and several different values of its inclination, this author was able to show that a region of stability only exists at distances of 0.3 to 0.8 AU from the system’s primary star. These results slightly overlap with those presented by Verrier and Evans (2006). Haghighipour (2006) also showed that the habitable zone of the primary at 3.05 to 3.7 AU is unstable. The focus of this paper is on the existence of possible additional planets in the γ Cephei system. The objects of interest are those with masses equal to those of Jupiter, Saturn and Neptune, on orbits beyond the location of the actual planet of the system. 2. STABILITY OF AN ADDITIONAL JUPITER-LIKE PLANET In order to find out whether the γ Cephei system can harbor an additional large planet, one has to study the dynamical stability of such an object at different distances from the primary star. However, it is important to choose the initial value of the semimajor axis of this planet to be outside the Hill sphere of other objects. The Hill sphere is the sphere of gravitational influence of an astronomical body in the face of the perturbations of the body it orbits around. One can calculate the Hill radius (radius of the Hill sphere) of a body from equation m 1/3 RH = a( ) . (1) 3M∗ In this equation, a is the semimajor axis of the object, m is its mass, and M∗ is the mass of the central body. In the case of γ Cephei, the Hill radius of the known Jupiter-like planet is approximately at 0.15 AU, and the Hill radius of the binary companion is at 8.1 AU (Verrier and Evans 2006). Taking into account the uncertainty on the semimajor axis of the known planet, and the furthest distance it can orbit due to its eccentricity, any additional planets must be placed at least 2.65 AU from the primary star. As for the binary, one can calculate its closest approach via the formula r = ab(1 − eb). (2) Given that ab=18.5 AU and eb=0.361, the value of r is approximately 12 AU. This means that at certain points in its orbit, the binary’s effect can stretch to less than 4 AU from the primary star. Any additional planets should technically not be placed any further out than 4 AU, but since the binary can influence at this distance only every 57 years (one period of its orbit), there is some leeway. Numerical integrations were carried out using the hybrid symplectic and also the Bulirsch-Stoer integrators of the MERCURY 6.2 Integrator Package (Chambers 1999). The timestep of each simulation was ten days. The orbital parameters of the known planet of the system and those of the binary were taken from Table 1. The initial longitude of ascending node (Ω), argument of pericentre (ω), and mean anomaly (M) of the additional planet were set to zero and the semimajor axis, eccentricity, mass and inclination of this planet were varied. Table 2 shows an overview of these values. Figure 1 shows the time-variations of the semimajor axis, eccentricity, and inclination of the three bodies of the system for one of our simulations. As shown here, the system becomes unstable after approximately 2 million years. It is important to note the jump in semimajor axis, eccentricity, and inclination of the binary and Jupiter-like planet after the additional planet becomes unstable and leaves the system. To conserve angular momentum, when the additional object leaves the system, the other two bodies move to different orbits. A more pronounced representation of this jump can be seen in Figure 2. The instability of the additional planet was observed in all our simulations. Figure 3 shows the results of numerical integrations performed for a Jupiter-mass planet with 0 eccentricity and no inclination.