# American Institute of Mathematical Sciences

June  2017, 10(2): 467-479. doi: 10.3934/krm.2017018

## Approximate explicit stationary solutions to a Vlasov equation for planetary rings

 Department of Mathematics and Computer Science, University of Catania, Viale A. Doria 6,95125 Catania, Italy

Received  January 2015 Revised  May 2016 Published  November 2016

In this paper we consider a Vlasov or collisionless Boltzmann equation describing the dynamics of planetary rings. We propose a simple physical model, where the particles of the rings move under the gravitational Newtonian potential of two primary bodies. We neglect the gravitational forces between the particles. We use a perturbative technique, which allows to find explicit solutions at the first order and approximate solutions at the second order, by solving a set of two linear ordinary differential equations.

Citation: Armando Majorana. Approximate explicit stationary solutions to a Vlasov equation for planetary rings. Kinetic & Related Models, 2017, 10 (2) : 467-479. doi: 10.3934/krm.2017018
##### References:

show all references

##### References:
The density of mass of $\Psi_{1}$
The density of mass of $\Psi_{2}$
 [1] Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 [2] Jean-Baptiste Caillau, Bilel Daoud, Joseph Gergaud. Discrete and differential homotopy in circular restricted three-body control. Conference Publications, 2011, 2011 (Special) : 229-239. doi: 10.3934/proc.2011.2011.229 [3] Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229 [4] Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057 [5] Hadia H. Selim, Juan L. G. Guirao, Elbaz I. Abouelmagd. Libration points in the restricted three-body problem: Euler angles, existence and stability. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 703-710. doi: 10.3934/dcdss.2019044 [6] Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074 [7] Frederic Gabern, Àngel Jorba, Philippe Robutel. On the accuracy of restricted three-body models for the Trojan motion. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 843-854. doi: 10.3934/dcds.2004.11.843 [8] Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519 [9] Elbaz I. Abouelmagd, Juan Luis García Guirao, Jaume Llibre. Periodic orbits for the perturbed planar circular restricted 3–body problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1007-1020. doi: 10.3934/dcdsb.2019003 [10] Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157 [11] Qunyao Yin, Shiqing Zhang. New periodic solutions for the circular restricted 3-body and 4-body problems. Communications on Pure & Applied Analysis, 2010, 9 (1) : 249-260. doi: 10.3934/cpaa.2010.9.249 [12] Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609 [13] Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299 [14] Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631 [15] Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158 [16] Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85 [17] Marcel Guardia, Tere M. Seara, Pau Martín, Lara Sabbagh. Oscillatory orbits in the restricted elliptic planar three body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 229-256. doi: 10.3934/dcds.2017009 [18] Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062 [19] Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 [20] Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987

2020 Impact Factor: 1.432