Abstract
The symplectic Wisdom-Holman map revolutionized long-term integrations of planetary systems. There is freedom in such methods of how to split the Hamiltonian and which coordinate system to employ, and several options have been proposed in the literature. These choices lead to different integration errors, which we study analytically and numerically. The Wisdom-Holman method in Jacobi coordinates and the method of Hernandez, H16, compare favorably and avoid problems of some of the other maps, such as incorrect center-of-mass position or truncation errors even in the one-planet case. We use H16 to compute the evolution of Pluto's orbital elements over 500 million years in a new calculation.
Highlights
Symplectic integrators first became popular in the 1990’s (Yoshida 1990; Channell & Scovel 1990) and they revolutionized our ability to study chaotic Hamiltonian dynamical systems for long times
For long term planetary studies such as investigating chaos in our solar system or stability of newly discovered planetary systems they have become the standard algorithms for solving the orbital ordinary differential equations
We have studied and done error analyses of the major planetary symplectic integrators in the literature
Summary
Symplectic integrators first became popular in the 1990’s (Yoshida 1990; Channell & Scovel 1990) and they revolutionized our ability to study chaotic Hamiltonian dynamical systems for long times. (Hairer et al 2006), where Herr is known as the error Hamiltonian and can be written as a power series in h, and n is the order of the integrator. An N-body problem described by linear first order ODE’s, an h that is a fraction, say 5%, of the shortest orbital period is sufficiently small (Wisdom 2015) in the absence of close encounters. The paper is organized as follows: in Section 2 we review preliminaries: first we review how to construct symplectic integrators using the operator splitting method and discuss the n-planet problem, a special type of N-body problem. Two popular methods for incorporating close encounters are the integrators of Chambers (1999), MERCURY and Duncan et al (1998), SyMBA: both are based on the Wisdom-Holman method in Democratic Heliocentric coordinates (Section 4.1). For all the tests in this paper, we use the Kepler solver of Wisdom & Hernandez (2015)
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