One-Dimensional Maps.- Two-Dimensional Maps.- Chaos.- Fractals.- Chaos in Two-Dimensional Maps.- Chaotic Attractors.- Differential Equations.- Periodic Orbits and Limit Sets.- Chaos in Differential Equations.- Stable Manifolds and Crises.- Bifurcations.- Cascades.- State Reconstruction from Data.

Chaos: An Introduction to Dynamical Systems

1. Introduction 2. Radial velocities 3. Astrometry 4. Timing 5. Microlensing 6. Transits 7. Imaging 8. Host stars 9. Brown dwarfs and free-floating planets 10. Formation and evolution 11. Interiors and atmospheres 12. The Solar System Appendixes References Index.

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The Exoplanet Handbook

We present IAS15, a 15th-order integrator to simulate gravitational dynamics. The integrator is based on a Gaus-Radau quadrature and can handle conservative as well as non-conservative forces. We develop a step-size control that can automatically choose an optimal timestep. The algorithm can handle close encounters and high-eccentricity orbits. The systematic errors are kept well below machine precision and long-term orbit integrations over 10 9 orbits show that IAS15 is optimal in the sense that it follows Brouwer’s law, i.e. the energy error behaves like a random walk. Our tests show that IAS15 is superior to a mixed-variable symplectic integrator (MVS) and other high-order integrators in both speed and accuracy. In fact, IAS15 preserves the symplecticity of Hamiltonian systems better than the commonly-used nominally symplectic integrators to which we compared it. We provide an open-source implementation of IAS15. The package comes with several easy-to-extend examples involving resonant planetary systems, Kozai-Lidov cycles, close encounters, radiation pressure, quadrupole moment, and generic damping functions that can, among other things, be used to simulate planet-disc interactions. Other non-conservative forces can be added easily.

ias15: a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits

The discovery of a new faint object in the outer solar system, 1992 QB1, moving beyond the orbit of Neptune is reported. It is suggested that the 1992 QB1 may represent the first detection of a member of the Kuiper belt (Edgworth, 1949; Kuiper, 1951), the hypothesized population of objects beyond Neptune and a possible source of the short-period comets, as suggested by Whipple (1964), Fernandez (1980), and Duncan et al. (1988).

Discovery of the candidate Kuiper belt object 1992 QB1

P-stable linear symmetric multistep methods for periodic initial-value problems

Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits

Simplified model solar systems with known observational errors are considered in conducting a dynamical search for dark mass and its minimum detectable amount, and in determining the significance of observed anomalies. The numerical analysis of the dynamical influence of dark mass on the orbits of outer planets and comets is presented in detail. Most conclusions presented are based on observations of the four giant planets where the observational errors in latitude and longitude are independent Gaussian variables with a standard deviation. Neptune's long orbital period cannot be predicted by modern ephemerides, and no evidence of dark mass is found in considering this planet. Studying the improvement in fit when observations are fitted to models that consider dark mass is found to be an efficient way to detect dark mass. Planet X must have a mass of more than about 10 times the minimum detectable mass to locate the hypothetical planet. It is suggested that the IRAS survey would have already located the Planet X if it is so massive and close that it dynamically influences the outer planets. Orbital residuals from comets are found to be more effective than those from planets in detecting the Kuiper belt. 35 refs.

Dynamical limits on dark mass in the outer solar system

In a self-gravitating system of point particles such as a spherical star cluster, small disturbances to an orbit grow exponentially on a time-scale comparable with the crossing time. The results of N-body integrations are therefore extremely sensitive to numerical errors: in practice it is almost impossible to follow orbits of individual particles accurately for more than a few crossing times. We demonstrate that numerical orbits in the gravitational N-body problem are often shadowed by true orbits for many crossing times. This result enhances our confidence in the use of N-body integrations to study the evolution of stellar systems.

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Gerald D. Quinlan

Papers

Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits

Dynamical limits on dark mass in the outer solar system

On the reliability of gravitational N-body integrations

Dynamical limits on dark mass in the outer solar system.