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 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 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 A Symplectic Integration Scheme that allows Close Encounters between Massive Bodies

1999 ◽  
Vol 172 ◽  
pp. 449-450
Author(s):  
J.E. Chambers
Keyword(s):  
Central Body ◽  
Symplectic Integrators ◽  
Integration Scheme ◽  
Original Form ◽  
Integration Step ◽  
Satellite Systems ◽  
Term Evolution ◽  
Two Bodies ◽  
Mixed Variable

Mixed-variable symplectic integrators provide a fast, moderately accurate way to study the long-term evolution of a wide variety of N-body systems (Wisdom & Holman 1991). They are especially suited to planetary and satellite systems, in which a central body contains most of the mass. However, in their original form, they become inaccurate whenever two bodies approach one another closely. Here, I will show how to overcome this difficulty using a hybrid integrator that combines symplectic and conventional algorithms.A symplectic integrator works by splitting the Hamiltonian, H, for an N-body system, into two or more parts H = H0 + H1 + …, where є i = Hi/H0 ≪ 1 for i = 1, 2 …. An integration step consists of several substeps, each of which advances the system due to the effect of one part of the Hamiltonian only. The error incurred over the whole step is ∼ є τn1, where τ is the timestep, n is the order of the integrator, and є is the largest of єi.

scholarly journals Flyby encounters between two planetary systems II: exploring the interactions of diverse planetary system architectures

2020 ◽  
Vol 496 (2) ◽  
pp. 1149-1165 ◽  
Author(s):  
Daohai Li ◽  
Alexander J Mustill ◽  
Melvyn B Davies
Keyword(s):  
Angular Momentum ◽  
Planetary System ◽  
Open Cluster ◽  
Planetary Systems ◽  
Giant Planets ◽  
Diverse Range ◽  
System Architectures ◽  
Earth Systems ◽  
Term Evolution

ABSTRACT Planetary systems formed in clusters may be subject to stellar encounter flybys. Here, we create a diverse range of representative planetary systems with different orbital scales and planets’ masses and examine encounters between them in a typical open cluster. We first explore the close-in multisuper Earth systems ≲0.1 au. They are resistant to flybys in that only ones inside a few au can destabilize a planet or break the resonance between such planets. But these systems may capture giant planets on to wide orbits from the intruding star during distant flybys. If so, the original close-in small planets’ orbits may be tilted together through Kozai–Lidov mechanism, forming a ‘cold’ system that is significantly inclined against the equator of the central host. Moving to the intermediately placed planets around solar-like stars, we find that the planets’ mass gradient governs the systems’ long-term evolution post-encounter: more massive planets have better chances to survive. Also, a system’s angular momentum deficit, a quantity describing how eccentric/inclined the orbits are, measured immediately after the encounter, closely relates to the longevity of the systems – whether or not and when the systems turn unstable in the ensuing evolution millions of years post-encounter. We compare the orbits of the surviving planets in the unstable systems through (1) the immediate consequence of the stellar fly or (2) internal interplanetary scattering long post-encounter and find that those for the former are systematically colder. Finally, we show that massive wide-orbit multiplanet systems like that of HR 8799 can be easily disrupted and encounters at a few hundreds of au suffice.

Link-Level Simulator for Wireless local Area Network

2018 ◽  
Vol 6 (7) ◽  
pp. 314
Author(s):  
Chaithra. H. U ◽  
Vani H.R
Keyword(s):  
Wireless Local Area Network ◽  
Local Area Network ◽  
Local Area Networks ◽  
Local Area ◽  
Long Term Evolution ◽  
Wireless Local Area Networks ◽  
Area Network ◽  
Network Simulator ◽  
Term Evolution

Now a days in Wireless Local Area Networks (WLANs) used in different fields because its well-suited simulator and higher flexibility. The concept of WLAN  with  advanced 5th Generation technologies, related to a Internet-of-Thing (IOT). In this project, representing the Network Simulator (NS-2) used linked-level simulators for Wireless Local Area Networks and still utilized IEEE 802.11g/n/ac with advanced IEEE 802.11ah/af technology. Realization of the whole Wireless Local Area Networking linked-level simulators inspired by the recognized Vienna Long Term Evolution- simulators. As a outcome, this is achieved to link together that simulator to detailed performances of Wireless Local Area Networking with Long Term Evolution, operated in the similar RF bands. From the advanced 5th Generation support cellular networking, such explore is main because different coexistences scenario can arise linking wireless communicating system to the ISM and UHF bands.

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