# High precision symplectic integrators for the Solar System

@article{Farrs2013HighPS, title={High precision symplectic integrators for the Solar System}, author={Ariadna Farr{\'e}s and Jacques Laskar and Sergio Blanes and Fernando Casas and Joseba Makazaga and Ander Murua}, journal={Celestial Mechanics and Dynamical Astronomy}, year={2013}, volume={116}, pages={141-174} }

Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term studies in the Solar System. These comparisons are made in Jacobi and heliocentric coordinates and the implementation of the algorithms is fully detailed for practical use. We conclude that high order integrators should be privileged, with a preference for the new $$(10,6,4)$$ method of Blanes et al… Expand

#### Figures and Tables from this paper

#### 48 Citations

A study of symplectic integrators for planetary system problems: error analysis and comparisons

- Physics
- 2017

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,… Expand

High-order regularised symplectic integrator for collisional planetary systems

- Physics
- Astronomy & Astrophysics
- 2019

We present a new mixed variable symplectic (MVS) integrator for planetary systems that fully resolves close encounters. The method is based on a time regularisation that allows keeping the stability… Expand

New families of symplectic splitting methods for numerical integration in dynamical astronomy

- Mathematics, Physics
- 2013

We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very… Expand

Symplectic propagators for the Kepler problem with time-dependent mass

- Mathematics
- Celestial Mechanics and Dynamical Astronomy
- 2019

New numerical integrators specifically designed for solving the two-body gravitational problem with a time-varying mass are presented. They can be seen as a generalization of commutator-free… Expand

Dynamic stability of the Solar System: Statistically inconclusive results from ensemble integrations

- Physics
- 2014

Due to the chaotic nature of the Solar System, the question of its long-term stability can only be answered in a statistical sense, for instance, based on numerical ensemble integrations of nearby… Expand

Dedicated symplectic integrators for rotation motions

- Physics
- Celestial Mechanics and Dynamical Astronomy
- 2019

We propose to use the properties of the Lie algebra of the angular momentum to build symplectic integrators dedicated to the Hamiltonian of the free rigid body. By introducing a dependence of the… Expand

Are long-term N-body simulations reliable?

- Physics
- 2020

$N$-body integrations are used to model a wide range of astrophysical dynamics, but they suffer from errors which make their orbits diverge exponentially in time from the correct orbits. Over long… Expand

Symplectic orbit propagation based on Deprit’s radial intermediary

- Computer Science
- Astrodynamics
- 2018

This work presents the development of symplectic integrators of different orders for spacecraft orbit propagation, and shows that these integrators are more accurate and substantially faster than Runge-Kutta-based methods. Expand

Variational and symplectic integrators for satellite relative orbit propagation including drag

- Physics
- 2018

Orbit propagation algorithms for satellite relative motion relying on Runge–Kutta integrators are non-symplectic—a situation that leads to incorrect global behavior and degraded accuracy. Thus,… Expand

New Integration Methods for Perturbed ODEs Based on Symplectic Implicit Runge–Kutta Schemes with Application to Solar System Simulations

- Mathematics, Computer Science
- J. Sci. Comput.
- 2018

A family of integrators, flow-composed implicit Runge–Kutta methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step of an implicitrunge–kutta (IRK) method applied to a transformed system, with potential application to long-term solar system simulations. Expand

#### References

SHOWING 1-10 OF 55 REFERENCES

Modern Integrations of Solar System Dynamics

- Geology
- 2002

▪ Abstract Until the early 1990s, numerical simulations of Solar System dynamics were done using accurate but slow integrators. The typical timescales were on the order of a million years, apart from… Expand

New families of symplectic splitting methods for numerical integration in dynamical astronomy

- Mathematics, Physics
- 2013

We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very… Expand

Symplectic maps for the N-body problem.

- Physics
- 1991

The present study generalizes the mapping method of Wisdom (1982) to encompass all gravitational n-body problems with a dominant central mass. The rationale for the generalized mapping method is… Expand

High order symplectic integrators for perturbed Hamiltonian systems

- Mathematics, Physics
- 2000

A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + εB was given in (McLachlan, 1995). We give here a constructive proof that for all… Expand

A hybrid symplectic integrator that permits close encounters between massive bodies

- Physics
- 1999

Mixed-variable symplectic integrators exhibit no long-term accumulation of energy error, beyond that owing to round-off, and they are substantially faster than conventional N-body algorithms. This… Expand

A numerical experiment on the chaotic behaviour of the Solar System

- Physics
- Nature
- 1989

LAPLACE and Lagrange made an essential contribution to the study of the stability of the Solar System by proving analytically that, to first order in the masses, inclinations and eccentricities of… Expand

Confirmation of resonant structure in the solar system

- Physics
- 1992

Abstract Using a semianalytical secular theory, Laskar (1989, Nature 338, 237–238) computed the orbits of the planets over 200 million years and found that their motion, and especially the motion of… Expand

Exhaustive Search of Symplectic Integrators using Computer Algebra

- Mathematics
- 1996

We find symplectic integrators using universal exponential identities or relations among formal Lie series. We give here general methods to compute such identities in a free Lie algebra. We recover… Expand

A Multiple Time Step Symplectic Algorithm for Integrating Close Encounters

- Physics
- 1998

We present a new symplectic algorithm that has the desirable properties of the sophisticated but highly efficient numerical algorithms known as mixed variable symplectic (MVS) methods and that, in… Expand

Analytical Framework in Poincare Variables for the Motion of the Solar System

- Physics
- 1991

The subject of this meeting is chaotic behaviour in celestial mechanics. It just means that we realize, one hundred years after Poincare that the solutions of our equations are complicated. In order… Expand