Symplectic integrators for long-term integrations in celestial mechanics

1991 ◽  
Vol 52 (3) ◽  
pp. 221-240 ◽  
Author(s):  
Brett Gladman ◽  
Martin Duncan ◽  
Jeff Candy
Keyword(s):  
Celestial Mechanics ◽  
Symplectic Integrators

scholarly journals Revisiting high-order Taylor methods for astrodynamics and celestial mechanics

2021 ◽  
Author(s):  
Francesco Biscani ◽  
Dario Izzo
Keyword(s):  
Celestial Mechanics ◽  
General Purpose ◽  
Symplectic Integrators ◽  
Conservation Of Energy ◽  
Domain Specific ◽  
Numerical Tests ◽  
The Earth ◽  
Taylor Methods ◽  
Speed And Accuracy

Abstract We present heyoka, a new, modern and general-purpose implementation of Taylor’s integration method for the numerical solution of ordinary differential equations. Detailed numerical tests focused on difficult high-precision gravitational problems in astrodynamics and celestial mechanics show how our general-purpose integrator is competitive with and often superior to state-of-the-art specialised symplectic and non-symplectic integrators in both speed and accuracy. In particular, we show how Taylor methods are capable of satisfying Brouwer’s law for the conservation of energy in long-term integrations of planetary systems over billions of dynamical timescales. We also show how close encounters are modelled accurately during simulations of the formation of the Kirkwood gaps and of Apophis’ 2029 close encounter with the Earth (where heyoka surpasses the speed and accuracy of domain-specific methods). heyoka can be used from both C++ and Python, and it is publicly available as an open-source project.

scholarly journals On the accuracy of symplectic integrators for secularly evolving planetary systems

2019 ◽  
Vol 490 (4) ◽  
pp. 5122-5133 ◽  
Author(s):  
Hanno Rein ◽  
Garett Brown ◽  
Daniel Tamayo
Keyword(s):  
Planetary Systems ◽  
Symplectic Integrators ◽  
Apparent Paradox ◽  
Energy Error ◽  
Evolving Systems ◽  
Specific Term ◽  
Term Evolution ◽  
Specific Choice ◽  
Shadow Hamiltonian

ABSTRACT Symplectic integrators have made it possible to study the long-term evolution of planetary systems with direct N-body simulations. In this paper we reassess the accuracy of such simulations by running a convergence test on 20 Myr integrations of the Solar System using various symplectic integrators. We find that the specific choice of metric for determining a simulation’s accuracy is important. Only looking at metrics related to integrals of motions such as the energy error can overestimate the accuracy of a method. As one specific example, we show that symplectic correctors do not improve the accuracy of secular frequencies compared to the standard Wisdom–Holman method without symplectic correctors, despite the fact that the energy error is three orders of magnitudes smaller. We present a framework to trace the origin of this apparent paradox to one term in the shadow Hamiltonian. Specifically, we find a term that leads to negligible contributions to the energy error but introduces non-oscillatory errors that result in artificial periastron precession. This term is the dominant error when determining secular frequencies of the system. We show that higher order symplectic methods such as the Wisdom–Holman method with a modified kernel or the SABAC family of integrators perform significantly better in secularly evolving systems because they remove this specific term.

scholarly journals 7. Mécanique Céleste (Celestial Mechanics)

1991 ◽  
Vol 21 (1) ◽  
pp. 15-27 ◽  
Author(s):  
Jacques Henrard
Keyword(s):  
Celestial Mechanics ◽  
Computer Technology ◽  
Reference Frames ◽  
Inertial Coordinate System ◽  
Small Bodies ◽  
Term Evolution ◽  
Dynamical Astronomy ◽  
Virginia Beach ◽  
Study Institute

During 1988–1990 Commission 7 has sponsored or co-sponsored several IAU conferences: Colloquium No. 109 “Application of Computer Technology to Dynamical Astronomy” (Gaithersburg, July 1988), Symposium No. 141 “Inertial Coordinate System on the Sky” (Pulkovo, October 1989), Colloquium No. 127 “Reference Frames” (Virginia Beach, October 1990), Colloquium No. 132 “Instability, Chaos and Predictability in Celestial Mechanics and Stellar Systems” (Delhi, October 1990). The colloquium No. 118 “Dynamics of Small Bodies in the Solar System” which was to be held in Nanjing in June 1989 had unfortunately to be postponed then cancelled. Other meetings of interest to the members of Commission 7 were the 2nd Alexander von Humbolt Colloquium on “Long Term Evolution of Planetary Systems” (Ramsau, March 1988), the Colloquium “Asteroids, Comets, Meteors III” (Uppsala, June 1989), the colloquium “Mécanique Céleste et Systèmes Hamiltoniens” (Luminy, May 1990) and the NATO Advanced Study Institute on “Predictability, Stability and Chaos in N-Body Dynamical Systems” (Cortina d’Ampezzo, August 1990).

scholarly journals In memory of Konstantin Vladislavovich Kholshevnikov

2021 ◽  
Author(s):  
L. L. Sokolov ◽  
◽  
V. Sh. Shaidulin ◽  
Keyword(s):  
Celestial Mechanics ◽  
State University ◽  
Educational Work ◽  
The World

Konstantin Vladislavovich Kolshevnikov (19.01.1939—10.01.2021) was Head of the Department of Celestial Mechanics of St. Petersburg State University, that ruled it for 50 years. He got many important results in science and taught many students who continue his work around the world. Konstantin Vladislavovich created unique courses in celestial mechanics, was engaged in educational work and earned many honorary awards and titles. Long-term chairman of the organizing committee and the jury of the student report competition of the Winter Astronomical School.

scholarly journals New Methods for Long-Time Numerical Integration of Planetary Orbits

1992 ◽  
Vol 152 ◽  
pp. 395-406 ◽  
Author(s):  
Hiroshi Kinoshita ◽  
Hiroshi Nakai
Keyword(s):  
Angular Momentum ◽  
Numerical Integration ◽  
Planetary System ◽  
Symplectic Integrators ◽  
Truncation Errors ◽  
New Methods ◽  
Planetary Orbits ◽  
Long Time ◽  
Discretization Errors

When planetary orbits are numerically integrated for a long time by conventional integrators, the most serious problem is secular errors in the energy and the angular momentum of the planetary system due to discretization (truncation) errors. The secular errors in the energy and the angular momentum mean that the semi-major axes, the eccentricities, and the inclinations of planetary orbits have a secular error which grows linearly with time. Recently symplectic integrators and linear symmetric multistep integrators are found not to produce the secular errors in the energy and the angular momentum due to the discretization errors. Here we describe briefly both methods and discuss favorable properties of these integrators for a long-term integration of planetary orbits.

7 - Mecanique Celeste (Celestial Mechanics)

1988 ◽  
Vol 20 (1) ◽  
pp. 15-28
Author(s):  
V. A. Brumberg ◽  
J. Henrard ◽  
Ju. V. Batrakov ◽  
K. B. Bhatnagar ◽  
J. Chapront ◽  
...  
Keyword(s):  
Solar System ◽  
Celestial Mechanics ◽  
Body Problem ◽  
Dynamical Behaviour ◽  
General Assembly ◽  
Computational Techniques ◽  
New Delhi ◽  
Study Institute ◽  
European Regional

During 1985-1987 Celestial Mechanics has been intensively developed in all its branches embracing physical bases, mathematical aspects, computational techniques and astronomical objectives. Commission 7 has organized three IAU conferences: Symposium No. 114 “Relativity in Celestial Mechanics and Astrometry” (Leningrad, May 1985), Colloquium No. 96 “The Few Body Problem” (Turku, June 1987) and Topical Session “Resonances in the Solar System” of the X-th European Regional Astronomy Meeting (Prague, August 1987). Members of the commission have broadly participated in the NATO Advanced Study Institute “Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems” (Cortina d’Ampezzo, August 1987) and some other international and regional conferences. Prospects of the actual celestial mechanics investigations have been discussed at a session of Commission 7 at the XIX-th IAU General Assembly (New Delhi, November 1985). Three papers dealing with the unsolved problems of celestial mechanics were primarily addressed to the rising generation of celestial mechanicians (V. A. Brumberg and J. Kovalevsky, CM. 39, 133, 1986; P.K. Seidelmann, CM. 39, 141, 1986).

scholarly journals High-order regularised symplectic integrator for collisional planetary systems

2019 ◽  
Vol 628 ◽  
pp. A32 ◽  
Author(s):  
Antoine C. Petit ◽  
Jacques Laskar ◽  
Gwenaël Boué ◽  
Mickaël Gastineau
Keyword(s):  
Hybrid Methods ◽  
Planetary Systems ◽  
High Order ◽  
Symplectic Integrators ◽  
Step Size ◽  
Time Step ◽  
Close Encounters ◽  
The Stability ◽  
Strong Scattering

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 properties of the symplectic integrators while also reducing the effective step size when two planets encounter. We used a high-order MVS scheme so that it was possible to integrate with large time-steps far away from close encounters. We show that this algorithm is able to resolve almost exact collisions (i.e. with a mutual separation of a fraction of the physical radius) while using the same time-step as in a weakly perturbed problem such as the solar system. We demonstrate the long-term behaviour in systems of six super-Earths that experience strong scattering for 50 kyr. We compare our algorithm to hybrid methods such as MERCURY and show that for an equivalent cost, we obtain better energy conservation.

scholarly journals Improving the accuracy of simulated chaotic N-body orbits using smoothness

2019 ◽  
Vol 490 (3) ◽  
pp. 4175-4182 ◽  
Author(s):  
David M Hernandez
Keyword(s):  
Hybrid Methods ◽  
Hamiltonian Dynamics ◽  
Computational Cost ◽  
Symplectic Integrators ◽  
Jacobi Constant ◽  
Planetary Orbits ◽  
Particle Separations ◽  
Differentiability Class ◽  
Infinite Differentiability

ABSTRACT Symplectic integrators are a foundation to the study of dynamical N-body phenomena, at scales ranging from planetary to cosmological. These integrators preserve the Poincaré invariants of Hamiltonian dynamics. The N-body Hamiltonian has another, perhaps overlooked, symmetry: it is smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than 0. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose DCO can be tuned easily. Hybrid methods are smooth, except at a finite number of phase space points. We study chaotic planetary orbits in a test considered by Wisdom. We find that increasing smoothness, at negligible extra computational cost in particular tests, improves the Jacobi constant error of the orbits by about 5 orders of magnitude in long-term simulations. The results from this work suggest that smoothness of the N-body Hamiltonian is a property worth preserving in simulations.

scholarly journals High-order symplectic integrators for planetary dynamics and their implementation in rebound

2019 ◽  
Vol 489 (4) ◽  
pp. 4632-4640 ◽  
Author(s):  
Hanno Rein ◽  
Daniel Tamayo ◽  
Garett Brown
Keyword(s):  
Large Time ◽  
Good Accuracy ◽  
Long Term Evolution ◽  
High Order ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Available N ◽  
Term Evolution ◽  
Planetary Dynamics

ABSTRACT Direct N-body simulations and symplectic integrators are effective tools to study the long-term evolution of planetary systems. The Wisdom–Holman (WH) integrator in particular has been used extensively in planetary dynamics as it allows for large time-steps at good accuracy. One can extend the WH method to achieve even higher accuracy using several different approaches. In this paper, we survey integrators developed by Wisdom et al., Laskar & Robutel, and Blanes et al. Since some of these methods are harder to implement and not as readily available to astronomers compared to the standard WH method, they are not used as often. This is somewhat unfortunate given that in typical simulations it is possible to improve the accuracy by up to six orders of magnitude (!) compared to the standard WH method without the need for any additional force evaluations. To change this, we implement a variety of high-order symplectic methods in the freely available N-body integrator rebound. In this paper, we catalogue these methods, discuss their differences, describe their error scalings, and benchmark their speed using our implementations.

scholarly journals Commission 7: Celestial Mechanics and Dynamical Astronomy: (Mecanique Celeste et Astronomie Dynamique)

2000 ◽  
Vol 24 (1) ◽  
pp. 11-20
Author(s):  
Claude Froeschlé ◽  
John Hadjidemetriou ◽  
R. Dvorak ◽  
S. Ferraz-Mello ◽  
T. Fukushima ◽  
...  
Keyword(s):  
Celestial Mechanics ◽  
Symplectic Integrators ◽  
Main Belt ◽  
The Past ◽  
Dynamical Astronomy ◽  
Potential Applications ◽  
Qualitative Models ◽  
Scientific Context ◽  
Quantitative Results ◽  
Transfer Mechanisms

Research in Celestial Mechanics, for the past three years, has mainly focused on the understanding of Chaos on all its aspects. The always larger number of potential applications (meteors, KBO, NEA, asteroids of the main belt but also exoplanets or galactic motions) and the development of new efficient tools, like the symplectic integrators, have allowed the passage from QUALITATIVE models (for example the transfer mechanisms) to real QUANTITATIVE results (like the calculation of lifetimes). This important step has contributed to (re)create collaborations between theoreticians and observers (for example, in the prediction of catastrophic impacts) and to situate the Celestial Mechanics in a wider scientific context.

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