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December  2013, 6(4): 701-714. doi: 10.3934/krm.2013.6.701

## Unstable galaxy models

 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 $E.$
Citation: Zhiyu Wang, Yan Guo, Zhiwu Lin, Pingwen Zhang. Unstable galaxy models. Kinetic and 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), 859-867. [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, Leningrad University, 1962. [3] J. Barnes, P. Hut and J. Goodman, Dynamical instabilities in spherical stellar systems, Astrophysical Journal, 300 (1986), 112-131. doi: 10.1086/163786. [4] P. Bartholomew, On the theory of stability of galaxies, Monthly Notices of the Royal Astronomical Society, 151 (1971), 333-350. [5] G. Bertin, Dynamics of Galaxies, Cambridge University Press, Cambridge, 2000. [6] J. Binney and S. Tremaine, Galactic Dynamics (2nd Edition), Princeton University Press, 2008. [7] J. P. Doremus, M. R. Feix and G. Baumann, Stability of encounterless spherical stellar systems, Phys. Rev. Letts, 26 (1971), 725-728. [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), 467-470. [9] A. Fridman and V. Polyachenko, Physics of Gravitating System Vol I, Springer-Verlag, 1984. [10] J. Goodman, An instability test for nonrotating galaxies, Astrophysical Journal, 329 (1988), 612-617. doi: 10.1086/166407. [11] Y. Guo and Z. Lin, Unstable and stable galaxy models, Commun. Math. Phys., 279 (2008), 789-813. doi: 10.1007/s00220-008-0439-z. [12] M. Henon, Numerical experiments on the stability of spherical stellar systems, Astronomy and Astrophysics, 24 (1973), 229-238. [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), 489-509. doi: 10.1007/s00220-007-0212-8. [14] H. Kandrup and J. F. Signet, A simple proof of dynamical stability for a class of spherical clusters, The Astrophys. J., 298 (1985), 27-33. doi: 10.1086/163586. [15] H. Kandrup, A stability criterion for any collisionless stellar equilibrium and some concrete applications thereof, Astrophysical Journal, 370 (1991), 312-317. [16] D. Merritt, Elliptical galaxy dynamics, The Publications of the Astronomical Society of the Pacific, 111 (1999), 129-168. [17] P. L. Palmer, Stability of Collisionless Stellar Systems: Mechanisms for the Dynamical Structure of Galaxies, Kluwer Academic Publishers, 1994. [18] J. Perez and J. Aly, Stability of spherical stellar systems - I. Analytical results, Monthly Notices of the Royal Astronomical Society, 280 (1996), 689-699. doi: 10.1093/mnras/280.3.689. [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), 737-745. [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), 363-380. doi: 10.1017/S0305004199004193.

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##### References:
 [1] V. A. Antonov, Remarks on the problem of stability in stellar dynamics, Soviet Astr. J., 4 (1960), 859-867. [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, Leningrad University, 1962. [3] J. Barnes, P. Hut and J. Goodman, Dynamical instabilities in spherical stellar systems, Astrophysical Journal, 300 (1986), 112-131. doi: 10.1086/163786. [4] P. Bartholomew, On the theory of stability of galaxies, Monthly Notices of the Royal Astronomical Society, 151 (1971), 333-350. [5] G. Bertin, Dynamics of Galaxies, Cambridge University Press, Cambridge, 2000. [6] J. Binney and S. Tremaine, Galactic Dynamics (2nd Edition), Princeton University Press, 2008. [7] J. P. Doremus, M. R. Feix and G. Baumann, Stability of encounterless spherical stellar systems, Phys. Rev. Letts, 26 (1971), 725-728. [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), 467-470. [9] A. Fridman and V. Polyachenko, Physics of Gravitating System Vol I, Springer-Verlag, 1984. [10] J. Goodman, An instability test for nonrotating galaxies, Astrophysical Journal, 329 (1988), 612-617. doi: 10.1086/166407. [11] Y. Guo and Z. Lin, Unstable and stable galaxy models, Commun. Math. Phys., 279 (2008), 789-813. doi: 10.1007/s00220-008-0439-z. [12] M. Henon, Numerical experiments on the stability of spherical stellar systems, Astronomy and Astrophysics, 24 (1973), 229-238. [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), 489-509. doi: 10.1007/s00220-007-0212-8. [14] H. Kandrup and J. F. Signet, A simple proof of dynamical stability for a class of spherical clusters, The Astrophys. J., 298 (1985), 27-33. doi: 10.1086/163586. [15] H. Kandrup, A stability criterion for any collisionless stellar equilibrium and some concrete applications thereof, Astrophysical Journal, 370 (1991), 312-317. [16] D. Merritt, Elliptical galaxy dynamics, The Publications of the Astronomical Society of the Pacific, 111 (1999), 129-168. [17] P. L. Palmer, Stability of Collisionless Stellar Systems: Mechanisms for the Dynamical Structure of Galaxies, Kluwer Academic Publishers, 1994. [18] J. Perez and J. Aly, Stability of spherical stellar systems - I. Analytical results, Monthly Notices of the Royal Astronomical Society, 280 (1996), 689-699. doi: 10.1093/mnras/280.3.689. [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), 737-745. [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), 363-380. doi: 10.1017/S0305004199004193.
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