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Stability of Rotating Gaseous Stars

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Abstract

We consider stability of rotating gaseous stars modeled by the Euler–Poisson system with general equation of states. When the angular velocity of the star is Rayleigh stable, we proved a sharp stability criterion for axi-symmetric perturbations. We also obtained estimates for the number of unstable modes and exponential trichotomy for the linearized Euler–Poisson system. By using this stability criterion, we proved that for a family of slowly rotating stars parameterized by the center density with fixed angular velocity, the turning point principle is not true. That is, unlike the case of non-rotating stars, the change of stability of the rotating stars does not occur at extrema points of the total mass. By contrast, we proved that the turning point principle is true for the family of slowly rotating stars with fixed angular momentum distribution. When the angular velocity is Rayleigh unstable, we proved linear instability of rotating stars. Moreover, we gave a complete description of the spectra and sharp growth estimates for the linearized Euler–Poisson system.

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References

  1. Auchmuty, G.: The global branching of rotating stars. Arch. Ration. Mech. Anal. 114(2), 179–193 (1991)

    MathSciNet  MATH  Google Scholar 

  2. Auchmuty, G., Beals, R.: Variational solutions of some nonlinear free boundary problems. Arch. Ration. Mech. Anal. 43, 255–271 (1971)

    MathSciNet  MATH  Google Scholar 

  3. Balinsky, A., Evans, W., Lewis, R.: The Analysis and Geometry of Hardy’s Inequality. Universitext. Springer, Berlin (2015)

    MATH  Google Scholar 

  4. Bisnovaty-Kogan, S.I., Blinnikov, G.S.: Static criteria for stability of arbitrarily rotating stars. Astron. Astrophys. 31(4), 391–404 (1974)

    ADS  Google Scholar 

  5. Caffarelli, L., Friedman, A.: The shape of axi-symmetric rotating fluid. J. Funct. Anal. 694(35), 109–142 (1980)

    MATH  Google Scholar 

  6. Chandrasekhar, S.: Introduction to the Stellar Structure. University of Chicago Press, Chicago (1939)

    MATH  Google Scholar 

  7. Chandrasekhar, S.: Ellipsoidal Figures of Equilibrium. Yale University Press, Yale (1969)

    MATH  Google Scholar 

  8. Chandrasekhar, S., Lebovitz, N.R.: The pulsations and the dynamical stability of gaseous masses in uniform rotation. Astrophys. J. 152(1), 267–291 (1968)

    ADS  Google Scholar 

  9. Chanillo, S., Li, Y.: On diameters of uniformly rotating stars. Commun. Math. Phys. 166(2), 417–430 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  10. Friedman, A., Turkington, B.: Asymptotic estimates for an axisymmetric rotating fluid. J. Funct. Anal. 37(2), 136–163 (1980)

    MathSciNet  MATH  Google Scholar 

  11. Friedman, A., Turkington, B.: Existence and dimensions of a rotating white dwarf. J. Differ. Equ. 42(3), 414–437 (1981)

    ADS  MathSciNet  MATH  Google Scholar 

  12. Friedman, J., Ipser, J., Sorkin, R.: Turning-point method for axisymmetric stability of rotating relativistic stars. Astrophys. J. 325, 722–724 (1988)

    ADS  Google Scholar 

  13. Hazelhurst, J.: The stabilizing effect of rotation. Astron. Astrophys. 219, 181–184 (1994)

    ADS  Google Scholar 

  14. Heilig, U.: On Lichtenstein’s analysis of rotating Newtonian stars. In Annales de l’IHP Physique théorique 60, 457–487 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Heinzle, J.M., Uggla, C.: Newtonian stellar models. Ann. Phys. 308, 18–61 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  16. Jang, J.: Nonlinear instability theory of Lane–Emden stars. Commun. Pure Appl. Math. 67, 1418–1465 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Jang, J., Makino, T.: On slowly rotating axisymmetric solutions of the Euler–Poisson equations. Arch. Ration. Mech. Anal. 225, 873–900 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Jang, J., Makino, T.: On rotating axisymmetric solutions of the Euler–Poisson equations. J. Differ. Equ. 266(7), 3942–3972 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  19. Jang, J., Makino, T.: Linearized analysis of barotropic perturbations around spherically symmetric gaseous stars governed by the Euler–Poisson equations. J. Math. Phys. 61(5), 051508 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  20. Jardetzky, W.S.: Theories of Figures of Celestial Bodies. Interscience Publishers, Geneva (1958)

    MATH  Google Scholar 

  21. Kähler, H.: Rotational effects on stellar structure and stability. Astron. Astrophys. 288, 191–203 (1994)

    ADS  Google Scholar 

  22. Kato, T.: Perturbation Theory for Linear Operators. Reprint of the 1980 Edition. Springer, Berlin (1995)

    Google Scholar 

  23. Lebovitz, N.R.: The effect of an arbitrary law of slow rotation on the oscillations and the stability of gaseous masses. Astrophys. J. 160, 701 (1970)

    ADS  MathSciNet  Google Scholar 

  24. Ledoux, P.: On the radial pulsation of gaseous stars. Astrophys. J. 102(2), 143 (1945)

    ADS  MathSciNet  Google Scholar 

  25. Li, Y.: On uniformly rotating stars. Arch. Ration. Mech. Anal. 115, 367–393 (1991)

    MathSciNet  MATH  Google Scholar 

  26. Lichtenstein, L.: Untersuchungen über die Gleichgewichtsfiguren rotierender Flüssigkeiten, deren Teilchen einander nach dem Newtonschen Gesetze anziehen. Math. Z. 36, 481–562 (1933)

    MathSciNet  MATH  Google Scholar 

  27. Lin, S.S.: Stability of gaseous stars in spherically symmetric motions. SIAM J. Math. Anal. 28(3), 539–569 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

  28. Lin, Z., Zeng, C.: Instability, index theorem, and exponential trichotomy for linear Hamiltonian PDEs. Mem. Am. Math. Soc. 275(1347), 136 (2022)

    MathSciNet  MATH  Google Scholar 

  29. Lin, Z., Zeng, C.: Separable Hamiltonian PDEs and turning point principle for stability of gaseous stars. Commun. Pure. Appl. Math. 75(11), 2511–2572 (2022)

    MathSciNet  MATH  Google Scholar 

  30. Luo, T., Smoller, J.: Rotating fluids with self-gravitation in bounded domains. Arch. Ration. Mech. Anal. 173(3), 345–377 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Luo, T., Smoller, J.: Nonlinear dynamical stability of Newtonian rotating and non-rotating white dwarfs and rotating supermassive stars. Commun. Math. Phys. 284(2), 425–457 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  32. Luo, T., Smoller, J.: Existence and non-linear stability of rotating star solutions of the compressible Euler–Poisson equations. Arch. Ration. Mech. Anal. 191(3), 447–496 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Luo, T., Smoller, J.: On the Euler–Poisson equations of self-gravitating compressible fluids. In: Nonlinear Conservation Laws and Applications, vol. 153. Springer, New York (2011)

  34. Ostriker, J.P., Mark, W.K.: Rapidly rotating stars. I. The self-consistent-field method. Astrophys. J. 151, 1075–1088 (1968)

    ADS  Google Scholar 

  35. Pringle, J.E., King, A.R.: Astrophysical Flows. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  36. Rayleigh, L.: On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 11, 57–70 (1880)

    MathSciNet  MATH  Google Scholar 

  37. Shapiro, S.L., Teukolsky, S.A.: Black Holes, White Dwarfs, and Neutron Stars. Willey, New York (1983)

    Google Scholar 

  38. Sidorov, K.A.: Influence of rotation and a binary companion on the frequency of the radial pulsations of a homogeneous star. Astrophysics 18(1), 90–96 (1982)

    ADS  Google Scholar 

  39. Sidorov, K.A.: Structure and oscillations of rotating polytropes. Astrophysics 17(4), 427–436 (1982)

    ADS  MathSciNet  Google Scholar 

  40. Stahler, S.W.: The equilibria of rotating isothermal clouds—part two—structure and dynamical stability. Astrophys. J. 268, 165–184 (1983)

    ADS  Google Scholar 

  41. Strauss, W.A., Wu, Y.: Steady states of rotating stars and galaxies. SIAM J. Math. Anal. 49(6), 4865–4914 (2017)

    MathSciNet  MATH  Google Scholar 

  42. Strauss, W.A., Wu, Y.: Rapidly rotating stars. Commun. Math. Phys. 368(2), 701–721 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  43. Strauss, W.A., Wu, Y.: Rapidly rotating white dwarfs. Nonlinearity 33(9), 4783–4798 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  44. Takami, K., Rezzolla, L., Yoshida, S.: A quasi-radial stability criterion for rotating relativistic stars. Mon. Not. R. Astron. Soc. 416(1), L1–L5 (2011)

    ADS  Google Scholar 

  45. Tassoul, J.L.: Theory of Rotating Stars. Princeton University Press, Princeton, NJ (1978)

    Google Scholar 

  46. Tassoul, J.L.: Stellar Rotation. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

Download references

Acknowledgements

This work is supported partly by the NSF Grants DMS-1715201 and DMS-2007457 (Lin) and the China Scholarship Council No. 201806310066 (Wang).

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Zhiwu Lin

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

    Yucong Wang

  3. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Yucong Wang

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  1. Zhiwu Lin
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  2. Yucong Wang
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Correspondence to Yucong Wang.

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Communicated by A. Ionescu.

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Lin, Z., Wang, Y. Stability of Rotating Gaseous Stars. Commun. Math. Phys. 402, 1725–1763 (2023). https://doi.org/10.1007/s00220-023-04763-0

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