Joseph Castro Mentor: Nader Haghighipour

Transcription

1 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. 23), 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 19. 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 25 and 65 AU; distances at which the gravitational effect of the farther companion is insignificant. When the separation is less than 1 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 5 light-years from Earth. This system has a 1.59 solar-mass K1 IV subgiant as its primary star, and a solar-mass red giant as its secondary. The semimajor axis and eccentricity of this binary are 18.5 ± 1.1 AU and.361 ±.23, respectively (Hatzes et al. 23). A Jupiterlike planet with at least 1.7 Jupiter-masses, orbits the system s primary star in an orbit with a semimajor axis of 2.13 ±.5 AU and an eccentricity of.12 ±.5 (Hatzes et al. 23). Table 1 shows the six orbital parameters of the γ Cephei system as reported by Hatzes et al. (23). 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 23, a study by Dvorak et al. (23) confirmed the dynamical stability of this planet for a period of approximately 1 8 years. The stability of this object was also confirmed separately by Haghighipour (26). 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 (26) found that the orbit of this planet is stable when the value of the binary eccentricity is smaller than.5, and the inclination of the planet is between and 4. Studies have also been carried out to determine whether additional planets and asteroids can exist within the γ Cephei system. Verrier and Evans (26) 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.5 to.75 AU, 1.5 to 1.15 AU, and also 1.2 to 1.3 AU. Around the secondary, they identified a region of stability at approximately.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. 23) Parameter Planet Binary a (AU) 2.13 ± ± 1.1 e.12 ± ±.23 i (deg) Ω (deg) ω (deg) 49.6 ± ± 1.2 M (deg)

2 2 TABLE 2 Parameter Known Planet Extra Planet Binary Mass 1.7 M J M J.4 M a (AU) e i (deg) -1 Ω (deg) ω (deg) M (deg) extending out from near 65 AU. Instead of using test particles, Haghighipour (26) simulated the dynamics of an actual Earth-like planet. Considering a range of.3 to 4. 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.3 to.8 AU from the system s primary star. These results slightly overlap with those presented by Verrier and Evans (26). Haghighipour (26) also showed that the habitable zone of the primary at 3.5 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 R H = a( m 3M ) 1/3. (1) 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.15 AU, and the Hill radius of the binary companion is at 8.1 AU (Verrier and Evans 26). 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 = a b (1 e b ). (2) Given that a b =18.5 AU and e b =.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 eccentricity and no inclination. As shown here, the longest lifetime of the Jupiter-mass planet is no more than 15, years, and corresponds to a distance of 3.5 AU. This increased stability at the distance of 3.5 AU from the primary star was also reported by Verrier and Evans (26). We also carried out a few simulations at 2.75 AU, 5.25 AU and 5.5 AU. As expected, results indicate that the additional planet became unstable within a few tens of thousands of years. It is expected that decreasing or increasing the semimajor axes will futher cut short the lifetime of the additional planet due to its venturing into the Hill sphere of the other planet or the binary. In addition to investigating the dynamical stability of the additional planet at various semimajor axes, the stability of the system for various initial eccentricities of the additional planet was also examined. The value of the planet s

3 3 Semimajor Axis (AU) Eccentricity Binary Known Planet Additional Planet Inclination (Deg.) Millions of Years Fig. 1. Plots of semimajor axis, eccentricity, and inclination versus time for a typical simulation. The initial semimajor axis of the additional planet is 3.5AU, its initial eccentricity is, and its initial inclination is Binary Star Eccentricity Additional Planet Known Planet Semimajor Axis (AU) Fig. 2. Eccentricity versus semimajor axis for the system of Figure 1.

4 4 1e+8 1e Initial semimajor axis (AU) Fig. 3. Lifetime of additional Jupiter-mass planet with eccentricity and inclination. Black indicates the additoinal planet, red indicates the known Jupiter-sized object, and green indicates the single case where the known planet left the system first. 3.5 AU 3.75 AU 4. AU 1e Initial eccentricity Fig. 4. Stability of the system as the initial eccentricity of the additional planet vaires. Other than eccentricity, no other eccentricity appears to be preferred over another for the entire range of simulations. The initial semimajor axis of the planet was 3.5 AU in all cases shown here. eccentricity were chosen to be,.5,.1,.15,. Simulations were carried out for each value of the semimajor axis with the aforementioned eccentricities. Figure 4 shows the lifetime of the additional planet in terms of its initial eccentricity at three semimajor axes of 3.5 AU, 3.75 AU, and 4. AU. As shown in this figure, orbital eccentricity can play an important role in the stablitiy a planet. For most of our cases, an eccentricity of zero seemed to be preferred, but sometimes an eccentricity of.5 could increase the additional planet s lifetime by a considerable amount. Some of our simulations, however, resulted in an increased lifetime by increasing the eccentricity beyond.5. In general, it appeared that the additional planet was least stable at eccentrcity, leaving the system within a few thousand years. It is expected that increasing the eccentricity further will cause the system to become unstable. We also simulated the dynamics of the system with and additional planet with a mass of 5 to 2.5 M J. The semimajor axis and eccentricity of this object were varied for each mass. Initially, it appeared that the lifetime of the additional planet increased as the mass decreased. Figure 5, however, shows that there is no apparent trend to the lifetime of this object. This figure considers the stability of various masses, in cirular orbits, and with an initial semimajor axes of 3.5 AU. In addition to varying the semimajor axis, eccentricity, and mass of the additional Jupiter-like planet, the planet s inclination was also varied from to 1, with a 2 increment. In these simulations, no stable orbits were found beyond 2 million years. Figure 6 shows the results of some simulations for a few masses in circular orbits, initially at 3.5 AU. Here, it seems, no inclination is preferred over another. The lifetime of the additional planet does not increase or decrease as a function of the initial inclination. Therefore, it is expected that inclinations beyond 1 will not produce better results. Since a 5 Jupiter-mass planet appeared to have the greatest probability of being stable,

5 5 1e+8 1e Mass of additional planet (Jupiter-masses) Fig. 5. Additional planet lifetimes when the mass was varied. The planet was given an initial semimajor axis of 3.5 AU and a circular orbit. The longest lifetime occured for a 5 M J planet, and the shorest lifetime occurued for a 1.75 M J planet..5 Jup. Mass 1.5 Jup. Mass 2.5 Jup. Mass 1e Inclination (degrees) Fig. 6. Additional planet lifetimes when the inclination was varied for three different masses. TABLE 3 Parameter Known Planet Extra Planet Binary Mass 1.7 M J.5-1. M S.4 M a (AU) e i (deg) -1 Ω (deg) ω (deg) M (deg) more inclinations were tried for that case. The results indicated that the system was unstable. 3. STABILITY OF AN ADDITIONAL SATURN-LIKE PLANET Simulations were also carried out for an additional Saturn-like planet in the γ Cephei system. Table 3 shows the orbital parameters used in these simulations. Numerical simulations were performed for a Saturn-mass object as well as a.5 Saturn-mass planet. The initial semimajor axis of each planet was varied between 3 and 5 AU, with an eccentricity ranging from. to. The inclination was not varied in these simulations, and was kept constant at. The results mirror those of the Jupiter-like planet. That is, no stable orbits were found and the longest lifetimes

6 6 1e+8 1e Initial semimajor axis (AU) Fig. 7. Lifetime of an additional Saturn-mass planet with eccentricity and inclination. Black indicates the additoinal planet, and red indicates the known Jupiter-sized object. The longest Saturn lifetime was found at 3.5 AU. TABLE 4 Parameter Known Planet Extra Planet Binary Mass 1.7 M J M N.4 M a (AU) e i (deg) -1 Ω (deg) ω (deg) M (deg) were at 3.5 AU from the primary star. Figure 7 shows how the planet s stability varies with semimajor axis. 4. STABILITY OF AN ADDITIONAL NEPTUNE-LIKE PLANET Neptune-like planets were also placed in the γ Cephei system. Numerical integrations were carried out for a 1. and 1.5 Neptune-mass object. The object was placed between 3 to 5 AU and was given an eccentricity between. and (Table 4). Similar to the simulations of the Saturn-like planet, no inclinations other than were used. Results from these integrations were very similar to those seen in the case of the Saturn-like planet. The only difference is that the Neptune-like planet actually appeared to be less stable than the Saturn-like object. Figures 8 and 9 show plots of the semimajor axes and eccentricities of γ Cephei system when a Neptune-mass planet is present. Note the periodicity in the graphs of the known planet of the system (Figure 8). This is due to the closest approach between the binary and the additional object. 5. CONCLUSION We presented the results of over 2 numerical integrations of the three body system of γ Cephei, assuming additional Jupiter to Neptune-sized planets. Simulations indicated that no such large bodies can exist in the system for a period larger than a few million years. Results also indicate that a region of increased lifetime exists at 3.5 AU from the primary star. The first part of this study focused on the possibility of the presence of additional Jupiter-like planets in the γ Cephei binary-planetary system. Previous studies report the possbility of terrestrial planets close to the primary star, as well as possible planets/asteroids orbiting the secondary star. However, this study, focused on the possibility of an extra object in the region between 3 to 5 AU from the primary star. Simulations were run for ficticious planets of masses of 5 to 2.5 Jupiter-masses, with eccentricities ranging from. to 2. and inclinations to 1. Numerical integrations resulted in no long-term stable orbits. Also, due to the complex dynamics of the system, no stability trends were found through mass, eccentricity, or inclination variations. After all integrations were complete, the longest stable orbit was found to be approximately 2,4, years corresponding to a 5 M J planet initially at 3.5 AU and with an initial eccentricity of zero and inclination of 1. Similar results followed for the study of possible Saturn- and Neptune-like planets in the system. No long-term stable orbits were found. Comparing the results from these two studies show that the Saturn-like object may have a lifetime greater than that of the Neptune-like body. Even so, it is not expected that smaller bodies, such as Uranus-like

7 7 Semimajor Axis (AU) Binary Additional Planet Known Planet 2.7 1E5 Years Fig. 8. Semimajor axis vs time for the binary, Jupiter-like planet, and a Neptune-like object. The initial semimajor axis of the Neptune-like planet was 3.5 AU, with an eccentricity of and inclination Binary Eccentricity Additional Planet Known Planet.1 1E5 Years Fig. 9. Eccentricity vs time for the binary, Jupiter-like planet, and Neptune-like planet. The initial semimajor axis of the Neptune-like planet was 3.5 AU, with an eccentricity of and inclination.

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