scholarly journals Erratum: Three dimensional phase structures of mean motion resonances

2020 ◽  
Vol 495 (1) ◽  
pp. 413-416
Author(s):  
Hanlun Lei
Keyword(s):  
Three Dimensional ◽  
Mean Motion Resonances ◽  
Mean Motion ◽  
Phase Structures

Kozai mechanism inside mean motion resonances in the three-dimensional phase space

2020 ◽  
Vol 493 (4) ◽  
pp. 5816-5824 ◽  
Author(s):  
Yi Qi ◽  
Anton de Ruiter
Keyword(s):  
Phase Space ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Resonant Coupling ◽  
Mean Motion Resonances ◽  
Mean Motion ◽  
3D Space ◽  
Test Particles ◽  
Dimensional Phase Space ◽  
Three Body

ABSTRACT In this paper, we investigate the Kozai mechanism inside the inclined mean motion resonance (MMR) through a three-dimensional (3D) phase space. The Hamiltonian approximation for both prograde and retrograde MMRs is established by a semi-analytical method. We pick Jupiter as the disturber and study the Kozai mechanism in the Sun–Jupiter circular restricted three-body problem. Kozai islands of the prograde and retrograde MMRs are found in the 3D phase space. Numerical integration demonstrates that the locus of the orbit on the Kozai island is bounded by the Kozai island in the 3D phase space, so the orbit is locked in the Kozai+MMR state. The study of the Kozai dynamics inside a retrograde 1:1 MMR indicates that Kozai islands in the 3D phase space are just a sufficient condition for the Kozai+MMR mechanism rather than a necessary condition. There is no Kozai island in the 3D space for the retrograde 1:1 MMR, but the resonant coupling of Kozai with the retrograde 1:1 MMR appears in the phase space. Finally, dynamical behaviours of the two test particles located on Kozai islands are demonstrated in the ephemeris model.

scholarly journals Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

2021 ◽  
Vol 9 (1) ◽  
pp. 76
Author(s):  
Duoc Nguyen ◽  
Niels Jacobsen ◽  
Dano Roelvink
Keyword(s):  
Surface Waves ◽  
Deep Water ◽  
Surf Zone ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Breaking Waves ◽  
Successful Implementation ◽  
Random Waves ◽  
Mean Motion ◽  
Generalized Lagrangian

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

scholarly journals Motion of dust in mean motion resonances with planets

2009 ◽  
Vol 103 (4) ◽  
pp. 343-364 ◽  
Author(s):  
Pavol Pástor ◽  
Jozef Klačka ◽  
Ladislav Kómar
Keyword(s):  
Mean Motion Resonances ◽  
Mean Motion
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