Archive for Rational Mechanics and Analysis

, Volume 224, Issue 3, pp 1127–1159 | Cite as

Mass–Radius Spirals for Steady State Families of the Vlasov–Poisson System

  • Tobias Ramming
  • Gerhard Rein


We consider spherically symmetric steady states of the Vlasov–Poisson system, which describe equilibrium configurations of galaxies or globular clusters. If the microscopic equation of state, i.e., the dependence of the steady state on the particle energy (and angular momentum) is fixed, a one-parameter family of such states is obtained. In the polytropic case the mass of the state along such a one-parameter family is a monotone function of its radius. We prove that for the King, Woolley–Dickens, and related models this mass–radius relation takes the form of a spiral.


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  1. 1.
    Andréasson H., Rein G.: A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein–Vlasov system. Class. Quantum Grav. 23, 3659–3677 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andréasson H., Rein G.: On the steady states of the spherically symmetric Einstein–Vlasov system. Class. Quantum Gravity 24, 1809–1832 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Batt J., Faltenbacher W., Horst E.: Stationary spherically symmetric models in stellar dynamics. Arch. Ration. Mech. Anal. 93, 159–183 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Binney J., Tremaine S.: Galactic Dynamics. Princeton University Press, Princeton (1987)zbMATHGoogle Scholar
  5. 5.
    Gidas B., Ni W.-M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guo Y., Rein G.: A non-variational approach to nonlinear stability in stellar dynamics applied to the King model. Commun. Math. Phys. 271, 489–509 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hadžić M., Rein G.: Stability for the spherically symmetric Einstein–Vlasov system—a coercivity estimate. Math. Proc. Camb. Philos. Soc. 155, 529–556 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hadžić M., Rein G.: On the small redshift limit of steady states of the spherically symmetric Einstein–Vlasov system and their stability. Math. Proc. Camb. Philos. Soc. 159, 529–546 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Heinzle J., Rendall A., Uggla C.: Theory of Newtonian self-gravitating stationary spherically symmetric systems. Math. Proc. Camb. Philos. Soc. 140, 177–192 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lemou M., Méhats F., Raphaël P.: Orbital stability of spherical galactic models. Invent. math. 187, 145–194 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Makino T.: On the spiral structure of the (R,M)-diagram for a stellar model of the Tolman–Oppenheimer–Volkoff equation. Funkcialaj Ekvacioj 43, 471–489 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ramming, T.: Über Familien sphärisch symmetrischer stationärer Lösungen des Vlasov–Poisson-Systems. PhD thesis, Bayreuth 2012Google Scholar
  13. 13.
    Ramming T., Rein G.: Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case—a simple proof for finite extension. SIAM J. Math. Anal. 45, 900–914 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rein G.: Reduction and a concentration-compactness principle for energy-Casimir functionals. SIAM J. Math. Anal. 33, 896–912 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rein G.: Non-linear stability of gaseous stars. Arch. Ration. Mech. Anal. 168, 115–130 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rein, G.: Collisionless kinetic equations from astrophysics—the Vlasov–Poisson system. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations, Evolutionary Equations, vol. 3. Elsevier, North Holland, 2007Google Scholar
  17. 17.
    Rein G.: Galactic dynamics in MOND—existence of equilibria with finite mass and compact support. Kinet. Relat. Models 8, 381–394 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rein G., Rendall A.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Philos. Soc. 128, 363–380 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sansone G.: Sulle soluzione die Emden dell’equazione di Fowler. Rend. Mat. Roma 1, 163–176 (1940)zbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Fakultät für Mathematik, Physik und InformatikUniversität BayreuthBayreuthGermany

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