The three-body problem

The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis, search for periodic orbits and resonances, and application of the results to some interesting astronomical and space dynamical settings. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications.Comment: 49 pages, 10 figures, Published in Reports on Progress in Physic

Mapping the Region in the Nearest Star System to Search for Habitable Planets

Circumstellar planets within the alpha Centauri AB star system have been suggested through formation models and recent observations, and ACESat (Belikov et al. AAS Meeting #225, #311.01, 2015) is a proposed space mission designed to directly image Earth-sized planets in the habitable zones of both of these stars. The alpha Centauri system is billions of years old, so planets are only expected to be found in regions where their orbits are long-lived. We evaluate the extent of the regions within the alpha Centauri AB star system where small planets are able to orbit for billion-year timescales and we map the positions in the sky plane where planets on stable orbits about either stellar component may appear. We confirm the qualitative results of Wiegert & Holman (Astron. J. 113, 1445, 1997) regarding the approximate size of the regions of stable orbits, which are larger for retrograde orbits relative to the binary than for prograde orbits. Additionally, we find that mean motion resonances with the binary orbit leave an imprint on the limits of orbital stability, and the effects of the Lidov-Kozai mechanism are also readily apparent. Overall, orbits in the habitable zones near the plane of the binary are stable, whereas high-inclination orbits are short-lived

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.Comment: 24 pages, 8 figures, 3 tables; International Journal of Astrobiology (in press

The Instability Transition for the Restricted 3-Body Problem. III. The Lyapunov Exponent Criterion

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 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. 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 of the binary components and the initial distance ratio of the planet relative to the stellar separation distance. Our theoretical results allow 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.Comment: 10 pages, 8 figures, 3 tables; accepted by Astronomy and Astrophysic

The Stability of the Suggested Planet in the nu Octantis System: A Numerical and Statistical Study

We provide a detailed theoretical study aimed at the observational finding about the nu Octantis binary system that indicates the possible existence of a Jupiter-type planet in this system. If a prograde planetary orbit is assumed, it has earlier been argued that the planet, if existing, should be located outside the zone of orbital stability. However, a previous study by Eberle & Cuntz (2010) [ApJ 721, L168] concludes that the planet is most likely stable if assumed to be in a retrograde orbit with respect to the secondary system component. In the present work, we significantly augment this study by taking into account the observationally deduced uncertainty ranges of the orbital parameters for the stellar components and the suggested planet. Furthermore, our study employs additional mathematical methods, which include monitoring the Jacobi constant, the zero velocity function, and the maximum Lyapunov exponent. We again find that the suggested planet is indeed possible if assumed to be in a retrograde orbit, but it is virtually impossible if assumed in a prograde orbit. Its existence is found to be consistent with the deduced system parameters of the binary components and of the suggested planet, including the associated uncertainty bars given by observations.Comment: 11 pages, 10 figures, 3 tables; Monthly Notices of the Royal Astronomical Society (in press

Treatment of secondary hyperparathyroidism in kidney disease: what we know and do not know about use of calcimimetics and vitamin D analogs

There is a growing understanding of the pathophysiology of secondary hyperparathyroidism (SHPT) and a recent emergence of new agents for SHPT treatment in patients with advanced kidney disease. At the same time, appreciation that mineral metabolic derangements promote vascular calcification and contribute to excess mortality, along with recognition of potentially important “non-classical” actions of vitamin D, have prompted the nephrology community to reexamine the use of various SHPT treatments, such as activated vitamin D sterols, phosphate binders, and calcimimetics. In this review, the evidence for treatment of SHPT with calcimimetics and vitamin D analogs is evaluated, with particular consideration given to recent clinical trials that have reported encouraging findings with cinacalcet use. Additionally, several controversies in the pathogenesis and treatment of SHPT are explored. The proposition that calcitriol deficiency is a true pathological state is challenged, the relative importance of the vitamin D receptor and the calcium sensing receptor in parathyroid gland function is summarized, and the potential relevance of non-classical actions of vitamin D for patients with advanced renal disease is examined. Taken collectively, the balance of evidence now supports a treatment paradigm in which calcimimetics are the most appropriate primary treatment for SHPT in the majority of end stage renal disease patients, but which nevertheless acknowledges an important role for modest doses of activated vitamin D sterols

Planetary Candidates Observed by Kepler V: Planet Sample from Q1-Q12 (36 Months)

The Kepler mission discovered 2842 exoplanet candidates with 2 years of data. We provide updates to the Kepler planet candidate sample based upon 3 years (Q1-Q12) of data. Through a series of tests to exclude false-positives, primarily caused by eclipsing binary stars and instrumental systematics, 855 additional planetary candidates have been discovered, bringing the total number known to 3697. We provide revised transit parameters and accompanying posterior distributions based on a Markov Chain Monte Carlo algorithm for the cumulative catalogue of Kepler Objects of Interest. There are now 130 candidates in the cumulative catalogue that receive less than twice the flux the Earth receives and more than 1100 have a radius less than 1.5 Rearth. There are now a dozen candidates meeting both criteria, roughly doubling the number of candidate Earth analogs. A majority of planetary candidates have a high probability of being bonafide planets, however, there are populations of likely false-positives. We discuss and suggest additional cuts that can be easily applied to the catalogue to produce a set of planetary candidates with good fidelity. The full catalogue is publicly available at the NASA Exoplanet Archive.Comment: Accepted for publication, ApJ

Kepler eclipsing binary stars. VII. the catalogue of eclipsing binaries found in the entire Kepler data set

The primary Kepler Mission provided nearly continuous monitoring of ~200,000 objects with unprecedented photometric precision. We present the final catalog of eclipsing binary systems within the 105 deg2 Kepler field of view. This release incorporates the full extent of the data from the primary mission (Q0-Q17 Data Release). As a result, new systems have been added, additional false positives have been removed, ephemerides and principal parameters have been recomputed, classifications have been revised to rely on analytical models, and eclipse timing variations have been computed for each system. We identify several classes of systems including those that exhibit tertiary eclipse events, systems that show clear evidence of additional bodies, heartbeat systems, systems with changing eclipse depths, and systems exhibiting only one eclipse event over the duration of the mission. We have updated the period and galactic latitude distribution diagrams and included a catalog completeness evaluation. The total number of identified eclipsing and ellipsoidal binary systems in the Kepler field of view has increased to 2878, 1.3% of all observed Kepler targets