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The Instability Transition for the Restricted 3-Body Problem. III. the Previous Papers in This Series

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The Instability Transition for the Restricted 3-Body Problem. III. the Previous Papers in This Series Astronomy & Astrophysics manuscript no. ms16199 c ESO 2018 September 8, 2018 The instability transition for the restricted 3-body problem III. The Lyapunov exponent criterion B. Quarles1, J. Eberle1, Z. E. Musielak1,2, and M. Cuntz1,3 1 Department of Physics, Science Hall, University of Texas at Arlington, Arlington, TX 76019-0059, USA; e-mail: [billyq,wjeberle,zmusielak,cuntz]@uta.edu 2 Kiepenheuer-Institut f¨ur Sonnenphysik, Sch¨oneckstr. 6, 79104 Freiburg, Germany 3 Institut f¨ur Theoretische Astrophysik, Universit¨at Heidelberg, Albert Uberle¨ Str. 2, 69120 Heidelberg, Germany Received ¡date¿ / Accepted ¡date¿ ABSTRACT 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. Key words. stars: binaries: general — celestial mechanics — chaos — stars: planetary systems 1. Introduction The study of planetary orbital stability is timely for arXiv:1106.5062v1 [astro-ph.SR] 24 Jun 2011 various astronomical reasons. First, although most plan- A classical problem within the realm of orbital stability ets are found in wide binaries, various cases of plan- studies for theoretical and observed planets in stellar binary ets in binaries with separation distances of less than systems is the circular restricted 3-body problem (CR3BP) 30 AU have also been identified (e.g., Patience et al. 2002; (e.g., Szebehely 1967; Roy 2005). The CR3BP describes Eggenberger et al. 2004; Eggenberger & Udry 2010, and the motion of a body of negligible1 mass moving in the references therein). Second, many more cases of planets in gravitational field of the two massive primaries considered stellar binary systems are expected to be discovered not- here to be two stars. The stars move in circular orbits about ing that binary (and possibly multiple) stellar systems oc- the center of mass and their motion is not influenced by the cur in high frequency in the local Galactic neighbourhood third body, the planet. Furthermore, the initial velocity of (Duquennoy & Mayor 1991; Lada 2006; Raghavan et al. the planet is assumed in the same direction as the orbital 2006). Moreover, orbital stability studies of planets around velocity of its host star, which is usually the more massive stars, including binary systems, are highly significant in of the two stars. consideration of ongoing and future planet search missions (e.g., Catanzarite et al. 2006; Cockell et al. 2009). Send offprint requests to: Z. E. Musielak The CR3BP has been studied in detail by 1 Negligible mass means that although the body’s motion is Dvorak (1984), Dvorak (1986), Rabl & Dvorak (1988), influenced by the gravity of the two massive primaries, its mass Smith & Szebehely (1993), Pilat-Lohinger & Dvorak is too low to affect the motions of the primaries. (2002), and Mardling (2007). It has also been the focus of 2 Quarles et al.: The instability transition for the restricted 3-body problem. III. the previous papers in this series. In Paper I, Eberle et al. 2. Theoretical approach (2008) obtained the planetary stability limits using a 2.1. Basic equations criterion based on Jacobi’s integral and Jacobi’s constant. The method, also related to the concept of Hill stability In the following, we consider the so-called coplanar CR3BP (Roy 2005), showed that orbital stability can be guaran- (e.g., Szebehely 1967; Roy 2005), which we will define as fol- teed only if the initial position of the planet lies within lows. Two stars are in circular motion about their common a well-defined limit determined by the mass ratio of the center of mass and their masses are much larger than the stellar components. Additionally, the stability criterion mass of the planet. In our case, the planetary mass is as- was found to be also related to the borders of the “zero sumed to be 1 × 10−6 of the mass of the star it orbits; also velocity contour” (ZVC) and its topology assessed by using note that the planetary motion is constrained to the orbital a synodic coordinate system regarding the two stellar plane of the two stars. Moreover, it is assumed that the ini- components. tial velocity of the planet is in the same direction as the In Paper II, Eberle & Cuntz (2010) followed another orbital velocity of its host star, which is the more massive theoretical concept based on the hodograph eccentricity cri- of the two stars, and that the starting position of the planet terion. This method relies on an approach given by differ- is to the right of the primary star (3 o’clock position), along ential geometry that analyzes the curvature of the plan- the line joining the binary components. etary orbit in the synodic coordinate system. The cen- Following the conventions described by Eberle et al. terpiece of this method is the evaluation of the effective (2008), we write the equations of motion in terms of the time-dependent eccentricity of the orbit based on the hodo- parameters µ and ρ0 with µ and α =1 − µ being related to graph in rotating coordinates as well as the calculation of the ratio of the two stellar masses m1 and m2 (see below). the mean and median values of the eccentricity distribu- Moreover, R0 denotes the initial distance of the planet from tion. This approach has been successful in mapping quasi- its host star, the more massive of the two stars with mass periodicity and multi-periodicity for planets in binary sys- m1, whereas D denotes the distance between the two stars, tems. It has also been tested by comparing its theoretical allowing us to define ρ0. In addition, we use a rotating refer- predictions with work by Holman & Wiegert (1999) and ence frame, which also gives rise to Coriolis and centrifugal Musielak et al. (2005) in regard to the extent of the region forces. The equations of motion are given as of orbital stability in binary systems of different mass ra- tios. x˙ = u (1) Previously, the work by Eberle & Cuntz (2010) dealt y˙ = v (2) with the assessment of short-time orbital stability, encom- z˙ = w (3) 3 passing time scales of 10 yrs or less. One of the findings x − µ x + α was the identification of a quasi-periodic region in the stel- u˙ = 2v + x − α − µ (4) r3 r3 lar mass and distance (µ, ρ0) parameter space (see Sect. 1 2 2.1). Due to the relatively short time scales considered in y y v˙ = −2u + y − α 3 − µ 3 (5) the previous study, there is a strong motivation to revisit r1 r2 the onset of orbital instability using longer time scales and z z w˙ = −α 3 − µ 3 (6) different types of methods. The focus of this paper is the r1 r2 analysis of orbital stability by Lyapunov exponents, which are among the most commonly used numerical tools for where m investigating chaotic behaviour of different dynamical sys- µ = 2 (7) tems (e.g., Hilborn 1994). The exponents have already re- m1 + m2 peatedly been used in orbital mechanics studies of the Solar α = 1 − µ (8) System (e.g., Lissauer 1999; Murray & Holman 2001). For 2 2 2 2 example, Lissauer (1999) discussed the long-term stability r1 = (x − µ) + y + z (9) 2 2 2 2 of the eight Solar System planets, while also considering r2 = (x + α) + y + z (10) previous studies. He concluded that the Solar System is R0 most likely astronomically stable, in the view of the limited ρ0 = (11) life time of the Sun; however, the orbits of Pluto and of D many asteroids may become unstable soon after the Sun The variables in the above equations describe the po- becomes a white dwarf.
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