Stability of P-type orbits around stellar binaries: An extension to counter-rotating orbits
Introduction
A new field of searches for exoplanets and exoplanetary systems is emerging since the discovery of the first extrasolar-planet orbiting the main-sequence star 51 Pegb (Mayor, Queloz, 1995, Lee, 2018). Various exoplanet databases are compiled by several groups, providing the scientific community improved understanding of planetary systems based on physical properties (Bashi et al., 2018), that continue to grow by, e.g., the recently launched Transiting Exoplanet Survey Satellite (TESS). Exploration of detailed properties such as atmospheric composition (Kempton, 2018) and spin (Snellen, 2014) of these exoplanets in their respective habitable zones (Kasting, et al., 1993, Kopparapu, Ramirez, 2013) will be possible by the upcoming advanced James Webb Space Telescope (JWST) and the Extremely Large Telescope (ELT).
Interestingly, a fair number, though small in percentage, of exoplanets are discovered about double stars systems. Traditionally, double star systems are considered in the canonical formation scenario of stellar formation, ensuring co-rotation of proto-planetary disks with the stellar binary. However, above mentioned advanced observatories promise a novel window to exoplanets also around binaries involving compact objects familiar from X-ray binaries (Imara and Di Stefano, 2018), whose formation history includes tidal capture (e.g. Ray et al., 1987).
In the model approximation of co-planar systems, exoplanets of the progenitors are hereby expected to appear on co- or counter-rotating orbits about the newly formed binary system with equal probability in the initial phase of tidal capture. This prospect may be seen more broadly in our quest to understand the formation and stability of circumbinary systems based, e.g., on ensemble statistics of their observed properties (Li et al., 2016). This is particularly prudent given the rich set of evolution scenarios on secular time scales, including flipping the orientation of an orbit by the Lidov–Kozai mechanism (Naoz, 2016) or evolution of the central binary (Munoz and Lai, 2015).
For a habitable zone to harbor exoplanets, it must allow orbits to be dynamically stable, providing sufficient time for life to develop (e.g. Eggl, Georgakarakos, Pilat-Lohinger, 2014, Westby, Conselice, 2020). For three-body systems, a common classification is in terms of P-type and S-type planetary orbits, where the first is circumbinary and second is around either one of the two primaries. The orbit is alternatively L-type, when the planet librates around one of the triangular Lagrangian points. The first detection of an S-type planetary orbit is in the Kepler system (Eggenberger, et al., 2004, Eggenberger, et al., 2007, Desidera, Barbieri, 2007, Roell, et al., 2012), followed by the P-type orbit of Kepler-16b (Doyle, 2011).
Stability of P-type orbits was first systematically studied by Dvorak (1986), presented in a stability diagram for the coplanar case as a function of eccentricity of an equal-mass central binary. Various extensions have been considered, e.g. finite mass ratios when planetary masses are small (Holman, Wiegert, 1999, Cuntz, 2015) and theoretical studies of orbital resonances (Morais and Giuppone, 2012). It is found that the inclination angle of the planetary orbit to orbital plane of the central binary, while observationally relevant, only weakly affects orbital stability (Pilat-Lohinger et al., 2003).
The habitable zone commonly refers to conditions facilitating the genesis and development of life on exoplanets. Habitability facilitating advanced life, however, may impose further conditions such as a global clement climate (van Putten, 2017, Lee, van Putten, 2019). As amply demonstrated by planets in our solar system, this requires relatively slow spin. If planets are born with arbitrary spin, this suggests the need for de-spinning by tidal interactions. In the habitable zone of the Sun, de-spinning is by lunar rather than the much weaker stellar tidal interactions (e.g. Fleming et al., 2018). Earth hereby successfully spun down to our present 24 h/day from 4.1 h/day initially over the course of its history. Relatively strong tidal interactions during the initial epoch of rapid spin (following birth of the Moon) was advantageous to biotic processes and abiogenesis (Lingam and Loeb, 2017). It gradually gave way to weak interactions today at slow spin as the Moon became more distant, advantageous to the formation of a global clement climate conducive to advanced life. Based on Earth’s history, therefore, a moon accompanying an exoplanet in the habitable zone may serve a proxy for conditions favoring advanced life.
In searches for advanced life on selected exoplanets by follow-up observations with upcoming advanced observatories, therefore, we are led to give detailed consideration to orbital stability of exoplanet - moon systems. To this end, we have build an accurate exosolar N-body simulator to handle a broad mass hierarchy defined by mass ratios of exoplanet to central binary and exomoon to exoplanet.
In this report, we highlight numerical results on the potential relevance of counter-rotating planetary obits around binaries involving a compact object, that may have formed by tidal capture. Revisiting stability of corotating P-type orbits following the pioneering study of Dvorak (1986), we quantify the change in the region of orbital stability for the counter-rotating case. In this process, we extend Dvorak’s original analysis of the three-body problem to include the unrestricted three-body problem, both for co- and counter-rotating orbits in the coplanar configuration for the case of equal mass central binaries with varying ellipticity.
For P-type orbits, Dvorak showed that a change of stability occurs across a gap between an outer and inner region of mostly stable and, respectively, unstable orbits, delineated by an Upper Critical Oribit (UCO) and, respectively, Lower Critical Orbit (LCO). UCO and LCO refers to the initial orbital radius of the third body, whose orbits are stable and, respectively, unstable after integration time over a fixed number of periods of the central binary. In between the UCO and LCO, results are ambiguous and sensitively depend on the choice of initial data, characteristic for the inherently chaotic behavior of three-body systems.
We set out to quantify the different behavior of UCO and LCO for the co- and counter rotation case. Our numerical results confirm and extend the stability diagram of Dvorak and are presented here also to serve as potential benchmarks for N-body simulators more generally. To this end, we pay specific attention to the problem of numerical convergence as a function of the number of orbital periods. Our model problem setting may thus serve as a starting problem for the much more general problem of non-coplanar orbits and non-equal masses.
Our exosolar system N-body simulator employs the ODE45 solver of MATLAB (MATLAB, 2017b). This is a fourth and fifth order adaptive ordinary differential equation solver. ODE45 performs accurate integration with numerical errors down to numerical round-off error in double precision of total-energy in the problem setting at hand, that focuses on a limited integration time. For very long integration time, (Zeebe, 2017) reviews many different methods for accurate integration that are outside the scope of this study. Having an accurate numerical solver at hand, we use a direct numerical integration in Cartesian coordinates, obviating the need for more advanced approaches such as corotating frames, otherwise well-known from numerical and analytical studies of disk instabilities (e.g. Goldreich, Goodman, Narayan, 1986, van Putten, 2002, D’Orazio, Haiman, Duffel, Mac Fayden, Farris, 2016). As a function of eccentricities from 0 to 0.9, we compute an extended stability diagram according to the following:
- 1.
Restricted and unrestricted three-body problem.In the three-body problem, the force of mutual attraction between the primaries and the third body is proportional to the mass of the latter. As the third body is taken to be small mass, it does not significantly influence the primaries. Neglicting this defines the restricted three-body problem. Stability is slightly perturbed in the un-restricted problem with small masses, that is readily resolved numerically at high resolution.
- 2.
Corotation and counter rotation.Since stability of P-type planetary orbits is expected to be sensitive to the angular velocity relative to the angular velocity of the central binary, corotation and counter rotation are expected to give different UCO and LCO, here studied side-by-side with otherwise the same set-up. In fact, since the angular velocity difference between that of the planet and the primary is relatively greater in the second case, we anticipate this second case to be relatively more stable;
- 3.
Convergence of numerical results against variation of integration time. We confirm modest sensitivity of UCO and LCO to orbital integration time.
Our roadmap is as follows. In Section 2, we discuss an initial set-up of restricted three-body problem including finite mass ratios of third body to study the stability of P-type orbits. Section 3, details the dynamical equations of motion. Results of P-type orbits stability are studied in Section 4. In Section 5, we discuss our results.
Section snippets
P-type orbits: initial conditions
To study dynamical stability of P-type orbits in the coplanar three-body problem, we set initial positions of three objects with dimensionless variables normalized to the solar system unit of length (1AU) and solar mass (M⊙ = 2 × 1033 g). The center of mass (CM) is at the origin (0,0,0) of a Cartesian coordinate system. Our equal mass primary binary has initial separation 1. The mass of the third body, m3 is of the total mass of the binary, .
In this configuration, closely following
Dynamical equations of motion
The general three-body problem for masses mi with position xi () and Newton’s constant G is described by the following the dynamical equations of motion in time t:
In our numerical implementation, we use dimensionless variables, i.e., masses are normalized to 1 solar mass and distances to 1 AU, as mentioned before, and . As such, time t is normalized to 1 yr.
P-type orbits: chaotic change of stability
Fig. 3 shows the overall increase of the UCO and LCO with e for the data in Fig. 2, along with the same for our counter rotationg case. The gap between the UCO and LCO is roughly uniform, apart from fluctuation inherent to chaotic behavior. The counter rotating case clearly lies below that of the corotating case. Stability extending closer in to the central binary, counter rotation is relatively more stable, even though it is more chaotic evidenced by a relatively more gradual transition
Conclusions
In light of the upcoming commissioning of advanced observatories, we consider the problem of stability of P-type orbits of exoplanets around mixed binaries involving compact objects. Familiar from studies of binary formation in dense stellar systems, these systems may form by tidal capture of a main sequence star. If so, exoplanets of such progenitor will be put on either co- or counter rotating orbits of the resulting binary with equal probability. This poses the novel question of stability of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors thank the anonymous reviewer for constructive comments. Support is acknowledged from the National Research Foundation of Korea under grants 2015R1D1A1A01059793, 2016R1A5A1013277 and 2018044640.
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