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 and Related Models, 2017, 10 (2) : 467-479. doi: 10.3934/krm.2017018
References:
[1] S. J. Aarseth, Gravitational N-Body Simulations: Tools and Algorithms, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511535246.
[2]

J. BattW. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Ration. Mech. Anal., 93 (1986), 159-183.  doi: 10.1007/BF00279958.

[3] J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, New York, 1988.  doi: 10.1063/1.2811635.
[4]

A. Bose and M. S. Janaki, Density distribution for an inhomogeneous finite gravitational system, Eur. Phys. J. B, 85 (2012), p360.  doi: 10.1140/epjb/e2012-30357-x.

[5] C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.  doi: 10.1007/978-1-4612-1039-9.
[6]

P.-H. Chavanis, Hamiltonian and Brownian systems with long-range interactions: Ⅰ Statistical equilibrium states and correlation functions, Physica A, 361 (2006), 55-80.  doi: 10.1016/j.physa.2005.06.087.

[7]

Y. Cheng and I. M. Gamba, Numerical study of one-dimensional Vlasov-Poisson equations for infinite homogeneous stellar systems, Nonlinear Sci. Numer. Simul., 17 (2012), 2052-2061.  doi: 10.1016/j.cnsns.2011.10.004.

[8]

P. Goldreich and S. Tremaine, The formation of the Cassini division in Saturn's rings, Icarus, 34 (1978), 240-253.  doi: 10.1016/0019-1035(78)90165-3.

[9]

E. GrivM. GedalinD. Eichler and C. Yuan, A gas-kinetic stability analysis of self-gravitating and collisional particulate disks with application to Saturn's rings, Planet. Space Sci., 48 (2000), 679-698.  doi: 10.1016/S0032-0633(00)00037-4.

[10]

E. GrivM. Gedalin and C. Yuan, On the stability of Saturn's rings: A quasi-linear kinetic theory, Mon. Not. R. Astron. Soc., 342 (2003), 1102-1116.  doi: 10.1046/j.1365-8711.2003.06608.x.

[11]

E. Griv and M. Gedalin, The fine-scale spiral structure of low and moderately high optical depth regions of Saturn's main rings: A review, Planet. Space Sci., 51 (2003), 899-927.  doi: 10.1016/j.pss.2003.05.003.

[12]

J. J. Lissauer and J. N. Cuzzi, Resonances in Saturn's rings, Astrophys. J., 87 (1982), 1051-1058.  doi: 10.1086/113189.

[13]

C. Mouhot, Stabilité orbitale pour le systéme de Vlasov-Poisson gravitationnel: (D'aprés Lemou-Méhats-Raphaël, Guo, Lin, Rein et al.), Asterisque, 352 (2013), 35-82. 

[14]

A. Ramírez-HernándezH. Larralde and F. Leyvraz, Violation of the zeroth law of thermodynamics in systems with negative specific heat, Phys. Rev. Lett., 100 (2008), 120601. 

[15]

G. Rein, Collisionless kinetic equations from astrophysics -the Vlasov-Poisson system, Handbook of differential equations: evolutionary equations, Elsevier/North-Holland, Amsterdam, 3 (2007), 383-476.  doi: 10.1016/S1874-5717(07)80008-9.

[16]

G. Severne and M. J. Haggerty, Kinetic theory for finite inhomogeneous gravitational systems, Astrophys. Space Sci., 45 (1976), 287-302.  doi: 10.1007/BF00642666.

[17] V. Szebehely, Theory of Orbit: The Restricted Problem of Three Bodies, Academic Press, New York and London, 1967. 
[18]

T. N. TelesY. LevinR. Pakter and F. B. Rizzato, Statistical mechanics of unbound two-dimensional self-gravitating systems, J. Stat. Mech., 2010 (2010), P05007.  doi: 10.1088/1742-5468/2010/05/P05007.

[19]

J. Touma and S. Tremaine, The statistical mechanics of self-gravitating Keplerian discs J. Phys. A: Math. Theor. 47 (2014), 292001, 25pp. doi: 10.1088/1751-8113/47/29/292001.

[20]

K. Yoshikawa, N. Yoshida and M. Umemura, Direct integration of the collisionless Boltzmann equation in six-dimensional phase space: Self-gravitating systems Astrophys. J. 762 (2013), art. no. 116. doi: 10.1088/0004-637X/762/2/116.

show all references

References:
[1] S. J. Aarseth, Gravitational N-Body Simulations: Tools and Algorithms, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511535246.
[2]

J. BattW. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Ration. Mech. Anal., 93 (1986), 159-183.  doi: 10.1007/BF00279958.

[3] J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, New York, 1988.  doi: 10.1063/1.2811635.
[4]

A. Bose and M. S. Janaki, Density distribution for an inhomogeneous finite gravitational system, Eur. Phys. J. B, 85 (2012), p360.  doi: 10.1140/epjb/e2012-30357-x.

[5] C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.  doi: 10.1007/978-1-4612-1039-9.
[6]

P.-H. Chavanis, Hamiltonian and Brownian systems with long-range interactions: Ⅰ Statistical equilibrium states and correlation functions, Physica A, 361 (2006), 55-80.  doi: 10.1016/j.physa.2005.06.087.

[7]

Y. Cheng and I. M. Gamba, Numerical study of one-dimensional Vlasov-Poisson equations for infinite homogeneous stellar systems, Nonlinear Sci. Numer. Simul., 17 (2012), 2052-2061.  doi: 10.1016/j.cnsns.2011.10.004.

[8]

P. Goldreich and S. Tremaine, The formation of the Cassini division in Saturn's rings, Icarus, 34 (1978), 240-253.  doi: 10.1016/0019-1035(78)90165-3.

[9]

E. GrivM. GedalinD. Eichler and C. Yuan, A gas-kinetic stability analysis of self-gravitating and collisional particulate disks with application to Saturn's rings, Planet. Space Sci., 48 (2000), 679-698.  doi: 10.1016/S0032-0633(00)00037-4.

[10]

E. GrivM. Gedalin and C. Yuan, On the stability of Saturn's rings: A quasi-linear kinetic theory, Mon. Not. R. Astron. Soc., 342 (2003), 1102-1116.  doi: 10.1046/j.1365-8711.2003.06608.x.

[11]

E. Griv and M. Gedalin, The fine-scale spiral structure of low and moderately high optical depth regions of Saturn's main rings: A review, Planet. Space Sci., 51 (2003), 899-927.  doi: 10.1016/j.pss.2003.05.003.

[12]

J. J. Lissauer and J. N. Cuzzi, Resonances in Saturn's rings, Astrophys. J., 87 (1982), 1051-1058.  doi: 10.1086/113189.

[13]

C. Mouhot, Stabilité orbitale pour le systéme de Vlasov-Poisson gravitationnel: (D'aprés Lemou-Méhats-Raphaël, Guo, Lin, Rein et al.), Asterisque, 352 (2013), 35-82. 

[14]

A. Ramírez-HernándezH. Larralde and F. Leyvraz, Violation of the zeroth law of thermodynamics in systems with negative specific heat, Phys. Rev. Lett., 100 (2008), 120601. 

[15]

G. Rein, Collisionless kinetic equations from astrophysics -the Vlasov-Poisson system, Handbook of differential equations: evolutionary equations, Elsevier/North-Holland, Amsterdam, 3 (2007), 383-476.  doi: 10.1016/S1874-5717(07)80008-9.

[16]

G. Severne and M. J. Haggerty, Kinetic theory for finite inhomogeneous gravitational systems, Astrophys. Space Sci., 45 (1976), 287-302.  doi: 10.1007/BF00642666.

[17] V. Szebehely, Theory of Orbit: The Restricted Problem of Three Bodies, Academic Press, New York and London, 1967. 
[18]

T. N. TelesY. LevinR. Pakter and F. B. Rizzato, Statistical mechanics of unbound two-dimensional self-gravitating systems, J. Stat. Mech., 2010 (2010), P05007.  doi: 10.1088/1742-5468/2010/05/P05007.

[19]

J. Touma and S. Tremaine, The statistical mechanics of self-gravitating Keplerian discs J. Phys. A: Math. Theor. 47 (2014), 292001, 25pp. doi: 10.1088/1751-8113/47/29/292001.

[20]

K. Yoshikawa, N. Yoshida and M. Umemura, Direct integration of the collisionless Boltzmann equation in six-dimensional phase space: Self-gravitating systems Astrophys. J. 762 (2013), art. no. 116. doi: 10.1088/0004-637X/762/2/116.

Figure 1.  The density of mass of $\Psi_{1}$
Figure 2.  The density of mass of $\Psi_{2}$
[1]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (182)
  • HTML views (47)
  • Cited by (0)

Other articles
by authors

[Back to Top]