American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation
  • KRM Home
  • This Issue
  • Next Article
    Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force
December  2013, 6(4): 701-714. doi: 10.3934/krm.2013.6.701

Unstable galaxy models

Zhiyu Wang 1, , Yan Guo 2, , Zhiwu Lin 3, and  Pingwen Zhang 1,

1. 

School of Mathematical Sciences, Peking University, Beijing, 100871, China, China
2. 

Division of Applied Mathematics, Brown University, Providence, RI 02912, United States
3. 

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States

Received  April 2013 Revised  June 2013 Published  November 2013

The dynamics of collisionless galaxy can be described by the Vlasov-Poisson system. By the Jean's theorem, all the spherically symmetric steady galaxy models are given by a distribution of $\Phi(E,L)$, where $E$ is the particle energy and $L$ the angular momentum. In a celebrated Doremus-Feix-Baumann Theorem [7], the galaxy model $\Phi(E,L)$ is stable if the distribution $\Phi$ is monotonically decreasing with respect to the particle energy $E.$ On the other hand, the stability of $\Phi(E,L)$ remains largely open otherwise. Based on a recent abstract instability criterion of Guo-Lin [11], we constuct examples of unstable galaxy models of $f(E,L)$ and $f\left( E\right) \ $in which $f$ fails to be monotone in $E.$
Keywords: Vlasov-Poisson, galaxy model., instability.
Mathematics Subject Classification: Primary: 85A05; Secondary: 74A2.
Citation: Zhiyu Wang, Yan Guo, Zhiwu Lin, Pingwen Zhang. Unstable galaxy models. Kinetic & Related Models, 2013, 6 (4) : 701-714. doi: 10.3934/krm.2013.6.701
References:
[1]

V. A. Antonov, Remarks on the problem of stability in stellar dynamics,, Soviet Astr. J., 4 (1960), 859. Google Scholar

[2]

V. A. Antonov, Solution of the Problem of Stability of Stellar System Emden'S Density Law and the Spherical Distribution of Velocities,, Vestnik Leningradskogo Universiteta, (1962). Google Scholar

[3]

J. Barnes, P. Hut and J. Goodman, Dynamical instabilities in spherical stellar systems,, Astrophysical Journal, 300 (1986), 112. doi: 10.1086/163786. Google Scholar

[4]

P. Bartholomew, On the theory of stability of galaxies,, Monthly Notices of the Royal Astronomical Society, 151 (1971), 333. Google Scholar

[5]

G. Bertin, Dynamics of Galaxies,, Cambridge University Press, (2000). Google Scholar

[6]

J. Binney and S. Tremaine, Galactic Dynamics (2nd Edition),, Princeton University Press, (2008). Google Scholar

[7]

J. P. Doremus, M. R. Feix and G. Baumann, Stability of encounterless spherical stellar systems,, Phys. Rev. Letts, 26 (1971), 725. Google Scholar

[8]

D. Gillon, M. Cantus, J. P. Doremus and G. Baumann, Stability of self-gravitating spherical systems in which phase space density is a function of energy and angular momentum, for spherical perturbations,, Astronomy and Astrophysics, 50 (1976), 467. Google Scholar

[9]

A. Fridman and V. Polyachenko, Physics of Gravitating System Vol I,, Springer-Verlag, (1984). Google Scholar

[10]

J. Goodman, An instability test for nonrotating galaxies,, Astrophysical Journal, 329 (1988), 612. doi: 10.1086/166407. Google Scholar

[11]

Y. Guo and Z. Lin, Unstable and stable galaxy models,, Commun. Math. Phys., 279 (2008), 789. doi: 10.1007/s00220-008-0439-z. Google Scholar

[12]

M. Henon, Numerical experiments on the stability of spherical stellar systems,, Astronomy and Astrophysics, 24 (1973), 229. Google Scholar

[13]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model,, Comm. Math. Phys., 271 (2007), 489. doi: 10.1007/s00220-007-0212-8. Google Scholar

[14]

H. Kandrup and J. F. Signet, A simple proof of dynamical stability for a class of spherical clusters,, The Astrophys. J., 298 (1985), 27. doi: 10.1086/163586. Google Scholar

[15]

H. Kandrup, A stability criterion for any collisionless stellar equilibrium and some concrete applications thereof,, Astrophysical Journal, 370 (1991), 312. Google Scholar

[16]

D. Merritt, Elliptical galaxy dynamics,, The Publications of the Astronomical Society of the Pacific, 111 (1999), 129. Google Scholar

[17]

P. L. Palmer, Stability of Collisionless Stellar Systems: Mechanisms for the Dynamical Structure of Galaxies,, Kluwer Academic Publishers, (1994). Google Scholar

[18]

J. Perez and J. Aly, Stability of spherical stellar systems - I. Analytical results,, Monthly Notices of the Royal Astronomical Society, 280 (1996), 689. doi: 10.1093/mnras/280.3.689. Google Scholar

[19]

J. F. Sygnet, G. des Forets, M. Lachieze-Rey and R. Pellat, Stability of gravitational systems and gravothermal catastrophe in astrophysics,, Astrophysical Journal, 276 (1984), 737. Google Scholar

[20]

G. Rein and A. D. Rendall, Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics,, Math. Proc. Camb Phil. Soc., 128 (2000), 363. doi: 10.1017/S0305004199004193. Google Scholar

show all references

References:
[1]

V. A. Antonov, Remarks on the problem of stability in stellar dynamics,, Soviet Astr. J., 4 (1960), 859. Google Scholar

[2]

V. A. Antonov, Solution of the Problem of Stability of Stellar System Emden'S Density Law and the Spherical Distribution of Velocities,, Vestnik Leningradskogo Universiteta, (1962). Google Scholar

[3]

J. Barnes, P. Hut and J. Goodman, Dynamical instabilities in spherical stellar systems,, Astrophysical Journal, 300 (1986), 112. doi: 10.1086/163786. Google Scholar

[4]

P. Bartholomew, On the theory of stability of galaxies,, Monthly Notices of the Royal Astronomical Society, 151 (1971), 333. Google Scholar

[5]

G. Bertin, Dynamics of Galaxies,, Cambridge University Press, (2000). Google Scholar

[6]

J. Binney and S. Tremaine, Galactic Dynamics (2nd Edition),, Princeton University Press, (2008). Google Scholar

[7]

J. P. Doremus, M. R. Feix and G. Baumann, Stability of encounterless spherical stellar systems,, Phys. Rev. Letts, 26 (1971), 725. Google Scholar

[8]

D. Gillon, M. Cantus, J. P. Doremus and G. Baumann, Stability of self-gravitating spherical systems in which phase space density is a function of energy and angular momentum, for spherical perturbations,, Astronomy and Astrophysics, 50 (1976), 467. Google Scholar

[9]

A. Fridman and V. Polyachenko, Physics of Gravitating System Vol I,, Springer-Verlag, (1984). Google Scholar

[10]

J. Goodman, An instability test for nonrotating galaxies,, Astrophysical Journal, 329 (1988), 612. doi: 10.1086/166407. Google Scholar

[11]

Y. Guo and Z. Lin, Unstable and stable galaxy models,, Commun. Math. Phys., 279 (2008), 789. doi: 10.1007/s00220-008-0439-z. Google Scholar

[12]

M. Henon, Numerical experiments on the stability of spherical stellar systems,, Astronomy and Astrophysics, 24 (1973), 229. Google Scholar

[13]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model,, Comm. Math. Phys., 271 (2007), 489. doi: 10.1007/s00220-007-0212-8. Google Scholar

[14]

H. Kandrup and J. F. Signet, A simple proof of dynamical stability for a class of spherical clusters,, The Astrophys. J., 298 (1985), 27. doi: 10.1086/163586. Google Scholar

[15]

H. Kandrup, A stability criterion for any collisionless stellar equilibrium and some concrete applications thereof,, Astrophysical Journal, 370 (1991), 312. Google Scholar

[16]

D. Merritt, Elliptical galaxy dynamics,, The Publications of the Astronomical Society of the Pacific, 111 (1999), 129. Google Scholar

[17]

P. L. Palmer, Stability of Collisionless Stellar Systems: Mechanisms for the Dynamical Structure of Galaxies,, Kluwer Academic Publishers, (1994). Google Scholar

[18]

J. Perez and J. Aly, Stability of spherical stellar systems - I. Analytical results,, Monthly Notices of the Royal Astronomical Society, 280 (1996), 689. doi: 10.1093/mnras/280.3.689. Google Scholar

[19]

J. F. Sygnet, G. des Forets, M. Lachieze-Rey and R. Pellat, Stability of gravitational systems and gravothermal catastrophe in astrophysics,, Astrophysical Journal, 276 (1984), 737. Google Scholar

[20]

G. Rein and A. D. Rendall, Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics,, Math. Proc. Camb Phil. Soc., 128 (2000), 363. doi: 10.1017/S0305004199004193. Google Scholar

[1]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
[2]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
[3]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic & Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729
[4]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[5]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[6]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[7]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039
[8]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046
[9]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[10]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic & Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011
[11]

Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723
[12]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283
[13]

Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic & Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729
[14]

Pablo Cincotta, Claudia Giordano, Juan C. Muzzio. Global dynamics in a self--consistent model of elliptical galaxy. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 439-454. doi: 10.3934/dcdsb.2008.10.439
[15]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479
[16]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707
[17]

Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems & Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163
[18]

Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457
[19]

Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681
[20]

Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences