Abstract
The traditional linearized stellar-structure equations, which provide information on the global stability of a stellar equilibrium state, are shown to engender a class of continuous modes, if the eigenfunctions are sought in a generalized function space. To endow the intuitive concept of ‘Local Stellar Statility’ with a precise mathematical meaning, we endeavour to link this notion to the family of continuous spectra. We indicate a simple numerical method to compute the eigenfunctions associated with these modes, and we illustrate our approach by briefly analysing the continuous spectra connected with the radial and nonradial dynamical stability problem.
Similar content being viewed by others
References
Aizenman, M. L. and Perdang, J.: 1973,Astron. Astrophys. 23, 209.
Aizenman, M. L. and Perdang, J.: 1976Astron. Astrophys. (in press).
Auré, J. L.: 1971,Astron. Astrophys. 11, 345.
Baker, N.: 1966, in A. G. W. Cameron and R. Stein (eds.),Stellar Evolution, Plenum Press, New York, p. 333.
Balescu, R.: 1975,Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley and Sons, New York, Chap. 3, 12.
Bouix, M.: 1964,Les Fonctions Généralisées ou Distributions, Masson et Cie, Paris, Chap. 8, 8.
Coddington, E. A. and Levinson, N.: 1955,Ordinary Differential Equations, McGraw-Hill Book Company, New York.
Courant, R. and Hilbert, D.: 1966,Methods of Mathematical Physics, Vol. II, Interscience Publishers, New York, Chap. VI.
Cristescu, R. and Marinescu, G.: 1973,Application of the Theory of Distributions, John Wiley and Sons, London.
Demaret, J.: 1975,Mém. Soc. Roy. Sc. Liège, 6e série,8, 161.
Doetsch, G.: 1937,Theorie und Anwendung der Laplace-Transformation, Springer, Berlin, 10. Kap.
Fjörtoft, R.: 1946,Geof. Pub. 16, No. 5.
Fricke, K. J. and Kippenhahn, R.: 1972,Ann. Rev. Astron. Astrophys. 10, 45.
Gabriel, M.: 1969,Astron. Astrophys. 1, 321.
Gantmacher, F. R.: 1966a,Théorie des Matrices Tome 1, Dunod, Paris, Chap. 5, 6.
Gantmacher, F. R.: 1966b,Théorie des Matrices, Tome 2, Dunod, Paris, Chap. 14.
Gel'fand, I. M. and Fomin, S. V.: 1963,Calculus of Variations, Prentice-Hall Inc., Englewood Cliffs, New Jersey, Sects. 25–30, 34.
Gel'fand, I. M. and Shilov, G. E.: 1964,Generalized Functions, Vol. I, Academic Press, New York, Chap. I.
Katz, A.: 1967,Principles of Statistical Mechanics, W. H. Freeman and Co., San Francisco, Chaps. 1, 5.
Kecs, W. and Teodorescu, P. P.: 1974,Applications of the Theory of Distributions in Mechanics, Abacus Press, Tunbridge Wells, Kent, Chaps. 1, 3.
Kharkevich, A. A.: 1960,Spectra and Analysis, Consultants Bureau, New York, Chap. 1, Sect. 12.
Kopal, Z.: 1948,Proc. Nat. Acad. Sci. U.S.A. 34, 377.
Landau, L. D. and Lifschitz, E. M.: 1966,Hydrodynamik, Akademie-Verlag, Berlin, Kap. III, § 28.
Ledoux, P.: 1958,Hdb. d. Phys. 51, 605.
Ledoux, P.: 1963,Acad. Roy. Belg. Bull. Cl. Sc., 5e série,49, 286.
Ledoux, P.: 1974, in P. Ledoux, A. Noels, and A. W. Rodgers (eds.), ‘Stellar Instability and Evolution’,IAU Symp. 59, 135.
Ledoux, P.: 1975,Stellar Stability, Symposium on Theoretical Astrophysics and Relativity, The University of Chicago, May 1975.
Ledoux, P. and Walraven, Th.: 1958,Hdb. d. Phys. 51, 353 (Sects. 73–83).
Lighthill, M. J.: 1958,Introduction to Fourier Analysis and Generalised Functions, Cambridge, University Press.
Perdang, J.: 1969, Ph.D. Thesis, Université de Liège.
Perdang J.: 1974, Lecture Notes, Istituto di Astronomia, Padova.
Perdang, J.: 1975,Adv. Chem. Phys. 32, 207.
Perdang, J.: 1976, Comptes-rendus du 94e Congrès de l'A.F.A.S., Bruxelles (in press).
Riesz, F. and Nagy, B. Sz.: 1953,Leçons d'Analyse Fonctionelle, Académie des Sciences de Hongrie, Budapest, Chap. 1, Sect. 7.
Robe, H. and Ledoux, P.: 1975,Acad. Roy. Belg. Bull. Cl. Sc. 5e série,61, 198.
Schwartz, L.: 1973,Théorie des distributions, Hermann, Paris, Chap. 5, § 6.
Simon, R.: 1975,Mém. Soc. Roy. Sc. Liège, 6e série,8, 57.
Smirnov, V. I.: 1964,A Course of Higher Mathematics, Vol. V, Pergamon Press, Oxford, Sect. 8.
Spiegel, E. A.: 1971,Ann. Rev. Astron. Astrophys. 9, 323.
Spiegel, E. A.: 1972,Ann. Rev. Astron. Astrophys. 10, 261.
Vogel, Th.: 1973,Pour une théorie mécaniste renouvelée, Gauthier-Villars, Paris, Chap. 12.
Wasiutyński, J.: 1946,Astrophysica Norvegica 4, 46.
Zahn, J. P.: 1974, in P. Ledoux, A. Noels, and A. W. Rodgers (eds.), ‘Stellar Instability and Evolution’,IAU Symp. 59, 185.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Perdang, J. Local stellar stability. Astrophys Space Sci 44, 177–215 (1976). https://doi.org/10.1007/BF00650482
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00650482