Dynamical Stability of Terrestrial and Giant Planets in the HD 155358 Planetary System

James Haynes

Advisor: Nader Haghighipour

ABSTRACT The results of a study of the dynamical evolution and the habitability of the planetary system of HD �155358 are presented. This system is unique in that it is one of the two low metallicity stars discovered to �host a multiple planet system. HD 155358 is host to two Jupiter-sized planets, with minimum masses of �0.86 and 0.50 Jupiter-masses. The orbit of the lower mass planet of this system is located at the inner �edge of the system's habitable zone. To determine whether this system can harbor terrestrial-type planets, �the orbits of its planets and an Earth-like object were numerically integrated for different values of their �masses and orbital eccentricities. Results indicate that this system could potentially host stable orbits for �terrestrial-sized planets in its habitable zone, but the stability of these orbits is very sensitive to the precise �characteristics of the giant planets of the system. The long-term stability of larger bodies (Neptune- and �Saturn-mass) was also studied in this system. Results show that a Neptune- or Saturn-mass body could �exist in stable orbits in this system fairly close in, potentially within the range of present detection �techniques, again depending on the precise configurations of the giant planets.

1. Introduction A survey of currently known extrasolar planet-hosting stars indicates that most of these stars have higher than solar metallicity (Figure 1). This is not unexpected since according to current theories of planet formation, planets grow from conglomeration of solid grains in circumstellar disks, implying that enhanced metallicity presumably implies more solid material in a planet-forming nebula, hence better chance for making planets. Some researchers have also argued that the presence of planets enriches their host stars with heavy elements (Pasquini et al., 2007). It is now evident that there is a correlation between the metallicity of a star and its capability in forming and harboring planets. Anomalies, however, exist among stars of extrasolar planetary systems. As shown in figure 1, there are two systems in which the central stars have metallicities lower than solar: HD 155358 and HD 1134729. These stars are hosts to giant planets, raising questions on how such planets have formed, whether these systems are dynamically stable, and whether terrestrial and/or habitable planets can exist in such systems. This paper focuses on seeking answers to these questions for the system of HD155358. Given that the second giant planet of this system lies inside its habitable zone, it would be interesting to investigate if this star can host additional objects. Here the possibility

Figure 1: Metallicity of extrasolar planet-hosting stars as a function of stellar mass. Most stars thus far found to host planets have super-solar levels of metal enrichment. A few stars including HD 155358 are, however, different. that additional planets, from Earth- to Saturn-mass, could exist in stable orbits in this system is examined. 2. The System: HD 155358 HD 155358 is a 10 Gyr old, 0.87 solar-masses G0 star at a distance of 42.7 ± 0.7 pc from the Sun. The effective temperature of this star is 5760 ± 101 K, and its metallicity, [Fe/H], is -0.68±0.07 solar. The luminosity of HD 155358 is approximately 1.545 times the luminosity of the Sun, which combined with the age of this star indicates that it is in fact a post-main sequence star (Cochran et al., 2007). This star is host to two giant planets with mass and orbital parameters given by (Cochran et al. 2007):

Planet HD 155358 b HD 155358 c Mass (M sin i) 0.89 ± 0.12 MJ 0.504 ± 0.075 MJ

Semimajor Axis a 0.628 ± 0.020 AU 1.224 ± 0.081 AU Eccentricity e 0.112 ± 0.037 0.176 ± 0.174 Argument of periastron ω 162 ± 20 ° 279 ± 38 ° Orbital Period P 195.0 ± 1.1 days 530 ± 27.2 days

Since the star HD 155358 is similar to the Sun, it is expected that its habitable zone will also lie in a region similar to that of the Sun. Defining a habitable zone as a region around a star where an Earth-like planet can retain liquid water on its surface for an extended period of time, it is in fact possible to calculate the boundaries of the habitable zone of HD 155358 by comparing its physical properties with those of the Sun. Given that the brightness of a star at the location of a planet plays the key role in planet’s capability in maintaining water on its surface and in its atmosphere, one can calculate the location of habitable zone by comparing a star’s luminosity with the solar luminosity using the equations

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where σ is the Boltzman’s constant, L is the star’s luminosity, R is its radius, T is its temperature, and r is the separation of the planet and the star. As shown by Kasting et al. (1993), the width of the habitable zone in our solar system ranges from 0.9 to 1.5 AU. However, subsequent work by Mischna et al. (2000) has indicated that this range is rather conservative, and that the true outer edge of Sun’s habitable zone may in fact be as far as 2.4 AU (Haghighipour, 2006). Using these values for the Sun’s habitable zone, and from equation (2), the habitable zone of HD 155358 extends from 1.18 to 1.99 AU, on the conservative estimate, or as far 2.98 AU in a more liberal estimate. In this study, the latter is indicated as an “extended” habitable zone. 3. Methodology To study the habitability of HD 155358, numerical integrations of the equations of motion of the two planets of this system and an Earth-like object at different distances from the star, were carried out using the hybrid routine of the N-body integration package MERCURY (Chambers 1999). This routine requires a timestep of equal to or shorter than 1/20th of the smallest period in the system. The inner planet of HD 155358 has a period of 195 days. A timestep of 9.75 days was used in the simulations. As mentioned earlier, the outer giant planet of the system is located near the inner edge of the system’s habitable zone. For this reason, additional planets were placed outside the outer edge of the influence zone of the second giant. The influence zone of a planetary body is defined as the region where the gravitational perturbation of the latter object will have pronounced effects on the motion of another body. The boundaries of this region, that are functions of planet’s mass, semimajor axis, and orbital eccentricity,

are given by Hill3)1( Rea +! and Hill3)1( Rea ++ . In this equation, RHill is the planet’s Hill radius, or region of gravitational dominance, and is defined thusly:

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Where a is the planet’s semimajor axis, Mp is the mass of the planet and M∗ is the mass of the central star (Menou and Tabachnik, 2002). Simulations were carried out for an object with a mass of 1, 5, 10, 20, and 100 Earth-masses in a coplanar orbit. The initial semimajor axis of this planet was chosen to be between 1.509 AU or 1.6 AU and 2.8 AU in steps of 0.1 AU. The other orbital parameters of this planet were assumed to be 0. Integrations were also carried out for an orbital inclination of 5 degrees for the Earth-size object. As the orbital inclinations of the two giant planes of the system with respect to the plane of the sky are unknown, simulations of the 1 Earth-mass body were repeated with the masses of the giant planets increased in order to reflect an orbital inclination of 45, 30, and 22 degrees. 4. Results a) Stability of an Earth-Mass Planet Results of simulations showed that, in general, the habitable zone of the system is unstable. Figure 2 shows the lifetime of an Earth-mass planet in this region. As shown in this figure, orbits become stable at larger distances (>2.6 AU), and the planet experiences small but periodic variations in eccentricity and semimajor axis. Figures 3 and 4 show these quantities for the Earth-mass object initially at 2.7 AU. Simulations indicated that the final results are sensitive to initial conditions. For instance, when the errors in the measurements of the orbital parameters of the giant planets of the system were taken into account, the orbit of an Earth-size planet became stable for the lowest reported values of these parameters. As shown in figure 5, the graph of the lifetimes contains an island of stability at 1.5 AU, which is very near both a 2:3 and 1:4 Mean Motion Resonance (MMR) with the outer and inner giant planets, respectively. This is within the conservative HZ of the system. Here, the orbit of the second giant planet becomes nearly circular (e = 0.02). Figure 6 shows the orbital parameters of an Earth-mass planet initially at 1.5 with the lowest values of the giants assumed. When the maximum values of the reported orbital parameters of the two giants were considered, the orbits of the Earth-size object was unstable for the entire habitable zone and also at larger distances (Figure 7). In 59% of cases the latter object collided with the central star or one of the giant planets, and in remaining cases it was ejected from the system.

Figure 2: Lifetime of an Earth-like planet vs its initial semi-major axis in the HD 155358 system. In all simulations up to 2.6 AU, orbit of the Earth-mass planet is unstable and the planet is ejected or collides with the giant planets or the star of the system. The HZ and the extended HZ for the system are also shown. The positions of giant planets are shown as red and orange dots, and the periastron to apastron distances of each planet are represented by horizontal lines going through the planet. The positions of Mean Motion Resonances are shown as vertical lines. The color of each MMR line represents the planet that the resonance corresponds to.

Figure 3: Graphs of the eccentricity and semi-major axes of the Earth-mass planet placed at an initial semimajor axis of 2.7 AU over the duration of the simulation. The periastron and apastron distances are shown on the lower graph in, colored in green and red, respectively. The eccentricity of the planet remains small and is comparable to Earth’s.

Figure 4: A closer view of the last 500,000 years of Figure 3. The eccentricity and semi-major axis of the Earth-like planet oscillate by a small amount in a very regular manner with a period of about 14,000 years.

Figure 5: Lifetime of an Earth-mass planet with the orbital parameters and masses of the giant planets assumed to be at their minimum measured values. The giants now have near circular orbits and the dynamics of the Earth-like planet in the HZ is chaotic. Resonances with giant planets now play a more important role in determining orbital stability.

Figure 6: Time variations of the semimajor axis and orbital eccentricity of the Earth-mass planet over the duration of the simulation with an initial semimajor axis of 1.5 AU. The giant planets were assumed to have the minimum values of the margin error associated with the literature mass, eccentricity and semi-major axis. The Earth-mass planet orbited very near a MMRs with each giant: 1:4 with the inner giant and 2:3 with the outer giant. The planet maintained a stable orbit despite fairly widely varying eccentricity.

Figure 7: Lifetime of an Earth-mass body assuming the maximum values of the measured orbital parameters of the second giant planet. The giant’s orbit becomes much more eccentric. No stable orbits are found. In every case the Earth-like planet is ejected from the system, or collides with one of the planets or the star within the duration of the simulation.

b) Stability of Additional Larger Bodies

Simulations using a 5 or 10 Earth-mass objects produced nearly identical results, as did simulations with a 20 Earth-mass body. They show that an object of up to Neptune-size remains stable at a distance as close as 2.6 AU. Figure 8 shows this in more detail for simulations with a 20 Earth-mass planet. Simulations with a 100 Earth-mass body, on the other hand, produced orbits that remained stable for 107 years at even closer distances. Results indicated that this object would remain stable for the duration of the integration as long as it does not approach the central star closer than 2.3 AU (figure 9). In this case, the orbits of all three planets become chaotic. It is possible that the orbit of a Saturn-mass object would prove to be ultimately unstable if the simulation was carried out beyond 107 years. Figure 10 shows the time variations of the semimajor axes of the system when a Saturn-mass body is initially at 2.4 AU from the central star. Simulations of the system with a 100 Earth-mass planet, and with the minimum and maximum measured values of the orbital parameters of the giant planets show that in the case of the minimum values, a Saturn-mass body is stable at 1.7 AU, whereas in the case of the maximum values, the entire system is stable for at least up to 107 years when the 100 Earth-mass body is placed at 3.3 AU. (Figures 10 and 11)

Figure 8: Lifetime of a 20 Earth-mass planet at different distances from the central star. The results are similar to a one Earth-mass object as shown in figure 2.

Figure 9: Lifetime of a 100 Earth-mass body. The stability is apparently enhanced, as this planet remains stable at closer distances.

Figure 10: Semimajor axes and periastron/apastron distances of all planets in the HD 155358 system over the duration of the simulation of a 100 Earth-mass object. The planets of the system undergo several temporary captures in MMRs. It is possible that these orbits may become unstable over longer time scales.

Figure 11: Lifetime of a 100 Earth-mass planet when numerically integrated with the maximum measured orbital parameters of the second giant. This configuration is not stable with the first two giant planets alone, however, islands of stability occur at the 1:4 and 1:5 MMR between the second giant and the Saturn-sized planet when this object is added to the system.

Figure 12: Lifetime of a 100 Earth-mass planet when numerically integrated with the lowest measured values for the orbital parameters of the giants. Here, most of the orbits are stable. Resonances play a destructive role, as in the case of the 3:4 and 3:2 MMR with the Saturn-like planet and the second giant. 1.7 AU is the minimum separation that produces stable orbits.

Figure 13: Lifetime of a 1 Earth-mass planet with the inclination of the entire system assumed to be 22 degrees. The masses of the giants have been scaled accordingly here. As shown here, there are no longer any stable orbits within the HZ for the Earth-size planet. c) Stability with Lesser System Inclination When the inclination of the system with respect to the plane of the sky was assumed to be 15 degrees, the masses of the giant planets increased to 3.3 and 1.96 MJ. In this case, the outer planet was ejected after 345,000 years, confirming the results reported by Cochran et al. (2007). As indicated by these authors, the system would be unstable when the orbital inclinations are less than 20 degrees. For inclinations of 22 degrees, the giant planets’ masses increase to 2.3 and 1.35 MJ. In this case, an island of stability appears at 3 AU, near a 1:4 MMR between the Earth-like planet and the outer giant, and just outside the most extended habitable zone. Figure 13 shows the graph of the lifetime of an Earth-mass object in this case. The results of simulations for inclinations of 30 and 45 degrees indicate that the Earth-size planet is stable only at 3.3 AU and 2.7 AU, respectively. The giant planets remain stable for the duration of the simulation in each case.

5. Conclusion

Results of the simulations of the stability of Earth-like and larger objects in the planetary system of HD 155358 were presented. Numerical integrations indicated that the habitable zone of this system is largely unstable, but these findings are highly sensitive to the initial conditions. The minimum distance where an Earth-like planet could orbit stably, is 2.6 AU, which is basically outside the system’s habitable zone. Simulations also indicated that, for lower values of the orbital inclinations of the giant planets of the system, an Earth-like planet could not exist in a stable orbit until much further out, outside the systems’ habitable zone Results also show that larger bodies, of Neptune- or Saturn-mass, could exist stably in this system in orbits as long as their radial distances to the central star do not exceed 2.6 AU and 2.3 AU, respectively. Our results also do not preclude the presence of additional planets orbiting inside the inner giant. Although simulations indicate that a terrestrial planet may have a stable orbit in this system, how such a planet could form is a question that requires more detailed study of terrestrial planet formation around low metallicity stars. It is uncertain whether in such systems there will be sufficient material to form such planets. Determining if such a planet can in fact exist around metal poor stars such as HD 155358 would be critical to our understanding of planet formation. This work has been supported by an NSF grant to the REU Program at the Institute for Astronomy at the University of Hawaii-Manoa. REFERENCES

Chambers, J.E. 1999, MNRAS, 304, 793 Cochran, W. D., Endl, M., Wittenmyer, R.A., and Bean, J.L. 2007, astroph/0705.3228 Haghighipour, N. 2006, ApJ, 644, 543 Menou, K. and Tabachnik, S. 2003 ApJ 583, 473-488 Pasquini, L., Döllinger M.P., Weiss, A., Girardi, L., Chavero C, Hatzes, A. P., da Silva, L., and Setiawan, J. 2007, astroph/0707.0788