J. Eberle

Stringent Criteria for Stable and Unstable Planetary Orbits in Stellar Binary Systems

The existence of planets in stellar binary (and higher order) systems has now been confirmed by many observations. The stability of planetary orbits in these systems has been extensively studied, but no precise stability criteria have so far been introduced. Therefore, there is an urgent need for developing stringent mathematical criteria that allow us to precisely determine whether a planetary orbit in a binary system is stable or unstable. In this Letter, such criteria are defined using the concept of Jacobis integral and Jacobis constant. These criteria are used to contest previous results on planetary orbital stability in binary systems.

ON THE REALITY OF THE SUGGESTED PLANET IN THE ν OCTANTIS SYSTEM

The aim of this study is to explore an enigmatic finding about the ? Octantis binary system that indicates the possible existence of a Jupiter-type planet even though the planet seems to be located outside the zone of orbital stability. We perform a detailed analysis of orbital stability based on previous studies that carefully considers the ? Octantis system parameters including their observationally deduced uncertainties. In our analysis, we confront the probability distribution of the parameter space of the system with the requirements of planetary orbital stability. Our results indicate that the suggested planet, if in a prograde orbit with respect to the motion of the binary components, is virtually impossible. However, the estimated probability of existence for a planet in a retrograde orbit is nearly 60%, an estimate that encapsulates the probability distribution of the mass ratio of the stellar components. This estimate increases if a relatively low stellar mass ratio (within the error bars) is assumed. The principal possibility of a planet in a retrograde orbit is also consistent with long-term orbital stability simulations pursued as part of our study. Thus, the existence of the suggested planet in the ? Octantis system constitutes a realistic possibility.

The instability transition for the restricted 3-body problem I. Theoretical approach

Aims. We study the onset of orbital instability for a small object, identified as a planet, that is part of a stellar binary system with properties equivalent to the restricted three body problem. Methods. Our study is based on both analytical and numerical means and makes use of a rotating (synodic) coordinate system keeping both binary stars at rest. This allows us to define a constant of motion (Jacobi’s constant), which is used to describe the permissible region of motion for the planet. We illustrate the transition to instability by depicting sets of time-dependent simulations with starplanet systems of different mass and distance ratios. Results. Our method utilizes the existence of an absolute stability limit. As the system parameters are varied, the permissible region of motion passes through the three collinear equilibrium points, which significantly changes the type of planetary orbit. Our simulations feature various illustrative examples of instability transitions. Conclusions. Our study allows us to identify systems of absolute stability, where the stability limit does not depend on the specifics or duration of time-dependent simulations. We also find evidence of a quasi-stability region, superimposed on the region of instability, where the planetary orbits show quasi-periodic behavior. The analytically deduced onset of instability is found to be consistent with the behavior of the depicted time-dependent models, although the manifestation of long-term orbital stability will require more detailed studies.

The instability transition for the restricted 3-body problem III. The Lyapunov exponent criterion

Aims. We establish a criterion for the stability of planetary orbits in stellar binary systems by using Lyapunov exponents and power spectra for the special case of the circular restricted 3-body problem (CR3BP). The criterion augments our earlier results given in the two previous papers of this series where stability criteria have been developed based on the Jacobi constant and the hodograph method. Methods. The centerpiece of our method is the concept of Lyapunov exponents, which are incorporated into the analysis of orbital stability by integrating the Jacobian of the CR3BP and orthogonalizing the tangent vectors via a well-established algorithm originally developed by Wolf et al. The criterion for orbital stability based on the Lyapunov exponents is independently verified by using power spectra. The obtained results are compared to results presented in the two previous papers of this series. Results. It is shown that the maximum Lyapunov exponent can be used as an indicator for chaotic behaviour of planetary orbits, which is consistent with previous applications of this method, particularly studies for the Solar System. The chaotic behaviour corresponds to either orbital stability or instability, and it depends solely on the mass ratio μ of the binary components and the initial distance ratio ρ0 of the planet relative to the stellar separation distance. Detailed case studies are presented for μ = 0.3 and 0.5. The stability limits are characterized based on the value of the maximum Lyapunov exponent. However, chaos theory as well as the concept of Lyapunov time prevents us from predicting exactly when the planet is ejected. Our method is also able to indicate evidence of quasi-periodicity. Conclusions. For different mass ratios of the stellar components, we are able to characterize stability limits for the CR3BP based on the value of the maximum Lyapunov exponent. This theoretical result allows us to link the study of planetary orbital stability to chaos theory noting that there is a large array of literature on the properties and significance of Lyapunov exponents. Although our results are given for the special case of the CR3BP, we expect that it may be possible to augment the proposed Lyapunov exponent criterion to studies of planets in generalized stellar binary systems, which is strongly motivated by existing observational results as well as results expected from ongoing and future planet search missions.

On the ejection of Earth-mass planets from the habitable zones of the solar twins HD 20782 and HD 188015

We provide a detailed statistical study of the ejection of fictitious Earth-mass planets from the habitable zones of the solar twins HD 20782 and HD 188015. These systems possess a giant planet that crosses into the stellar habitable zone, thus effectively thwarting the possibility of habitable terrestrial planets. In the case of HD 188015, the orbit of the giant planet is essentially circular, whereas in the case of HD 20782, it is extremely elliptical. As starting positions for the giant planets, we consider both the apogee and perigee positions, whereas the starting positions of the Earth-mass planets are widely varied. For the giant planets, we consider models based on their minimum masses as well as models where the masses are increased by 30%. Our simulations indicate a large range of statistical properties concerning the ejection of the Earth-mass planets from the stellar habitable zones. For example, it is found that the ejection times for the Earth-mass planets from the habitable zones of HD 20782 and HD 188015, originally placed at the centre of the habitable zones, vary by a factor of ~200 and ~1500, respectively, depending on the starting positions of the giant and terrestrial planets. If the mass of the giant planet is increased by 30%, the variation in ejection time for HD 188015 increases to a factor of ~6000. However, the short survival times of any Earth-mass planets in these systems are of no surprise. It is noteworthy, however, that considerable differences in the survival times of the Earth-mass planets are found, which may be relevant for establishing guidelines of stability for systems with less intrusive giant planets.

On the Possibility of Habitable Moons in the System of HD 23079: Results from Orbital Stability Studies

The aim of our study is to investigate the possibility of habitable moons orbiting the giant planet HD 23079b, a Jupiter-mass planet, which follows a low-eccentricity orbit in the outer region of HD 23079s habitable zone. We show that HD 23079b is able to host habitable moons in prograde and retrograde orbits, as expected, noting that the outer stability limit for retrograde orbits is increased by nearly 90% compared to that of prograde orbits, a result consistent with previous generalized studies. For the targeted parameter space it was found that the outer stability limit for habitable moons varies between 0.05236 and 0.06955 AU (prograde orbits) and between 0.1023 and 0.1190 AU (retrograde orbits) depending on the orbital parameters of the Jupiter-type planet if a minimum mass is assumed. These intervals correspond to 0.306 and 0.345 (prograde orbits) and 0.583 and 0.611 (retrograde orbits) of the planets Hill radius. Larger stability limits are obtained if an increased value for the planetary mass m_p is considered; they are consistent with the theoretically deduced relationship of m_p^{1/3}. Finally, we compare our results to the statistical formulae of Domingos et al. (2006) [MNRAS 373, 1227], indicating both concurrence and limitations.

Case studies of habitable Trojan planets in the system of HD 23079

We investigate the possibility of habitable Trojan planets in the HD 23079 star–planet system. This system consists of a solar-type star and a Jupiter-type planet, which orbits the star near the outer edge of the stellar habitable zone in an orbit of low eccentricity. We find that in agreement with previous studies Earth-mass habitable Trojan planets are possible in this system, although the success of staying within the zone of habitability is significantly affected by the orbital parameters of the giant planet and by the initial condition of the theoretical Earth-mass planet. In one of our simulations, the Earth-mass planet is captured by the giant planet and thus becomes a habitable moon.

The instability transition for the restricted 3-body problem - II. The hodograph eccentricity criterion

Aims. We present a new method that allows identifying the onset of orbital instability, as well as quasi-periodicity and multi-periodicity, for planets in binary systems. This method is given for the special case of the circular restricted 3-body problem (CR3BP). Methods. Our method relies on an approach given by differential geometry that analyzes the curvature of the planetary orbit in the synodic coordinate system. The centerpiece of the method consists in inspecting the effective (instantaneous) eccentricity of the orbit based on the hodograph in rotated coordinates and in calculating the mean and median values of the eccentricity distribution. Results. Orbital stability and instability can be mapped by numerically inspecting the hodograph and/or the effective eccentricity of the orbit in the synodic coordinate system. The behavior of the system depends solely on the mass ratio μ of the binary components and the initial distance ratio ρ 0 of the planet relative to the stellar separation distance noting that the stellar components move on circular orbits. Our study indicates that orbital instability occurs when the median of the effective eccentricity distribution exceeds unity. This instability criterion can be compared to other criteria, including those based on Jacobis integral and the zero-velocity contour of the planetary orbit. Conclusions. The method can be used during detailed numerical simulations and in contrast to other methods such as methods based on the Lyapunov exponent does not require a piece-wise secondary integration of the planetary orbit. Although the method has been deduced for the CR3BP, it is likely that it can be expanded to more general cases.

Orbital stability of Earth-type planets in stellar binary systems

An important factor in estimating the likelihood of life elsewhere in the Universe is determining the stability of a planets orbit. A significant fraction of stars like the Sun occur in binary systems which often has a considerable effect on the stability of any planets in such a system. In an effort to determine the stability of planets in binary star systems, we conducted a numerical simulation survey of several mass ratios and initial conditions. We then estimated the stability of the planetary orbit using a method that utilizes the hodograph to determine the effective eccentricity of the planetary orbit. We found that this method can serve as an orbital stability criterion for the planet.

Orbital stability of planets in binary systems: A new look at old results

About half of all known stellar systems with Sun-like stars consist of two or more stars, significantly affecting the orbital stability of any planet in these systems. This observational evidence has prompted a large array of theoretical research, including the derivation of mathematically stringent criteria for the orbital stability of planets in stellar binary systems, valid for the “coplanar circular restricted three-body problem”. In the following, we use these criteria to explore the validity of results from previous theoretical studies.