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

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

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[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. Google Scholar

[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. Google Scholar

show all references

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

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

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[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. Google Scholar

[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. Google Scholar

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