Approximate explicit stationary solutions to a Vlasov equation for planetary rings

Armando Majorana

doi:10.3934/krm.2017018

Article Contents

2017, Volume 10, Issue 2: 467-479. Doi: 10.3934/krm.2017018

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Approximate explicit stationary solutions to a Vlasov equation for planetary rings

Armando Majorana

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

Received: January 2015

Revised: May 2016

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q83; Secondary: 35Q85.

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

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

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References

	S. J. Aarseth, Gravitational N-Body Simulations: Tools and Algorithms, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511535246.
	J. Batt , W. Faltenbacher and E. Horst , Stationary spherically symmetric models in stellar dynamics, Ration. Mech. Anal., 93 (1986) , 159-183. doi: 10.1007/BF00279958.
	J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, New York, 1988. doi: 10.1063/1.2811635.
	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.
	C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.
	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.
	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.
	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.
	E. Griv , M. Gedalin , D. 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.
	E. Griv , M. 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.
	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.
	J. J. Lissauer and J. N. Cuzzi , Resonances in Saturn's rings, Astrophys. J., 87 (1982) , 1051-1058. doi: 10.1086/113189.
	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.
	A. Ramírez-Hernández , H. Larralde and F. Leyvraz , Violation of the zeroth law of thermodynamics in systems with negative specific heat, Phys. Rev. Lett., 100 (2008) , 120601.
	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.
	G. Severne and M. J. Haggerty , Kinetic theory for finite inhomogeneous gravitational systems, Astrophys. Space Sci., 45 (1976) , 287-302. doi: 10.1007/BF00642666.
	V. Szebehely, Theory of Orbit: The Restricted Problem of Three Bodies, Academic Press, New York and London, 1967.
	T. N. Teles , Y. Levin , R. 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.
	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.
	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.