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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

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