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The Oscillation Modes: Linear Perturbation Scheme

  • Pantelis PnigourasEmail author
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Part of the Springer Theses book series (Springer Theses)

Abstract

Starting with the standard equations of hydrodynamics (Sect. 2.1), we are going to derive the linear perturbation formalism (Sect. 2.2), used to obtain the oscillation modes of a star. We will first consider the simple case of a nonrotating star (Sect. 2.3), in which the various classes of modes are defined (Sect. 2.4), namely polar and axial modes. The former class contains the (fundamental) f-modes, the (pressure) p-modes, and the (buoyancy) g-modes, and the latter comprises the (rotational) r-modes. An additional class of modes, called hybrid modes, will also be discussed. For the sake of completeness, we provide the analytic formulae for the eigenfrequencies and eigenfunctions of the modes of a homogeneous star (Sect. 2.5), even though they are not used throughout this study. Finally, we consider the general case of a rotating star (Sect. 2.6), making use of the slow-rotation approximation, i.e., introducing (up to second-order) rotational corrections to the eigenfrequencies.

References

  1. Abramowitz, M., & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover. http://adsabs.harvard.edu/abs/1972hmfw.book.....A.
  2. Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. (2010). Asteroseismology. Astronomy and Astrophysics Library. New York: Springer. http://adsabs.harvard.edu/abs/2010aste.book.....A.CrossRefGoogle Scholar
  3. Ballot, J., Lignières, F., Reese, D. R., & Rieutord, M. (2010). Gravity modes in rapidly rotating stars. Limits of perturbative methods. Astronomy & Astrophysics, 518, A30.  https://doi.org/10.1051/0004-6361/201014426, arXiv:1005.0275.ADSCrossRefGoogle Scholar
  4. Chandrasekhar, S. (1963). Letter to the editor: A general variational principle governing the radial and the non-radial oscillations of gaseous masses. The Astrophysical Journal, 138, 896.  https://doi.org/10.1086/147694.ADSCrossRefGoogle Scholar
  5. Chandrasekhar, S. (1964). A general variational principle governing the radial and the non-radial oscillations of gaseous masses. The Astrophysical Journal, 139, 664.  https://doi.org/10.1086/147792.ADSMathSciNetCrossRefGoogle Scholar
  6. Chandrasekhar, S., & Lebovitz, N. R. (1964). Non-radial oscillations of gaseous masses. The Astrophysical Journal, 140, 1517.  https://doi.org/10.1086/148056.ADSCrossRefGoogle Scholar
  7. Clement, M. J. (1964). A general variational principle governing the oscillations of a rotating gaseous mass. The Astrophysical Journal, 140, 1045.  https://doi.org/10.1086/148004.ADSMathSciNetCrossRefGoogle Scholar
  8. Clement, M. J. (1965). The radial and non-radial oscillations of slowly rotating gaseous masses. The Astrophysical Journal, 141, 210.  https://doi.org/10.1086/148104.ADSMathSciNetCrossRefGoogle Scholar
  9. Cowling, T. G. (1941). The non-radial oscillations of polytropic stars. Monthly Notices of the Royal Astronomical Society, 101, 367.  https://doi.org/10.1093/mnras/101.8.367.ADSMathSciNetCrossRefGoogle Scholar
  10. Cunha, M. S. et al. (2007). Asteroseismology and interferometry. The Astronomy and Astrophysics Review, 14, 217–360.  https://doi.org/10.1007/s00159-007-0007-0, arXiv:0709.4613.ADSCrossRefGoogle Scholar
  11. Detweiler, S. L. (1975). A variational calculation of the fundamental frequencies of quadrupole pulsation of fluid spheres in general relativity. The Astrophysical Journal, 197, 203–217.  https://doi.org/10.1086/153504.ADSCrossRefGoogle Scholar
  12. Detweiler, S., & Lindblom, L. (1985). On the nonradial pulsations of general relativistic stellar models. The Astrophysical Journal, 292, 12–15.  https://doi.org/10.1086/163127.ADSCrossRefGoogle Scholar
  13. Dyson, J., & Schutz, B. F. (1979). Perturbations and stability of rotating stars. I. Completeness of normal modes. Proceedings of the Royal Society of London. Series A, 368, 389–410.  https://doi.org/10.1098/rspa.1979.0137.ADSMathSciNetCrossRefGoogle Scholar
  14. Dziembowski, W. (2010). Asteroseismology of rapidly rotating pulsators. Highlights of Astronomy, 15, 360–361.  https://doi.org/10.1017/S1743921310009804.CrossRefGoogle Scholar
  15. Dziembowski, W., & Goode, P. R. (1984). Simple asymptotic estimates of the fine structure in the spectrum of solar oscillations due to rotation and magnetism. Memorie della Societa Astronomica Italiana, 55, 185–213. http://adsabs.harvard.edu/abs/1984MmSAI.55.185D.
  16. Ferrari, V., Miniutti, G., & Pons, J. A. (2003). Gravitational waves from newly born, hot neutron stars. Monthly Notices of the Royal Astronomical Society, 342, 629–638.  https://doi.org/10.1046/j.1365-8711.2003.06580.x, arXiv:astro-ph/0210581.ADSCrossRefGoogle Scholar
  17. Ferrari, V., Gualtieri, L., Pons, J. A., & Stavridis, A. (2004). Rotational effects on the oscillation frequencies of newly born proto-neutron stars. Monthly Notices of the Royal Astronomical Society, 350, 763–768.  https://doi.org/10.1111/j.1365-2966.2004.07698.x, arXiv:astro-ph/0310896.ADSCrossRefGoogle Scholar
  18. Finn, L. S. (1986). g-modes of non-radially pulsating relativistic stars: The slow-motion formalism. Monthly Notices of the Royal Astronomical Society, 222, 393–416. http://adsabs.harvard.edu/abs/1986MNRAS.222.393F.ADSCrossRefGoogle Scholar
  19. Finn, L. S. (1987). G-modes in zero-temperature neutron stars. Monthly Notices of the Royal Astronomical Society, 227, 265–293. http://adsabs.harvard.edu/abs/1987MNRAS.227.265F.ADSCrossRefGoogle Scholar
  20. Friedman, J. L., & Stergioulas, N. (2013). Rotating relativistic stars. Cambridge, England: Cambridge University Press. http://adsabs.harvard.edu/abs/2013rrs..book.....F.
  21. Gaertig, E., & Kokkotas, K. D. (2008). Oscillations of rapidly rotating relativistic stars. Physical Review D, 78, 064063.  https://doi.org/10.1103/PhysRevD.78.064063, arXiv:0809.0629.
  22. Gough, D. O., & Taylor, P. P. (1984). Influence of rotation and magnetic fields on stellar oscillation eigenfrequencies. Memorie della Societa Astronomica Italiana, 55, 215–226. http://adsabs.harvard.edu/abs/1984MmSAI.55.215G.
  23. Gualtieri, L., Pons, J. A., & Miniutti, G. (2004). Nonadiabatic oscillations of compact stars in general relativity. Physical Review D, 70, 084009.  https://doi.org/10.1103/PhysRevD.70.084009, arXiv:gr-qc/0405063.
  24. Gualtieri, L., Kantor, E. M., Gusakov, M. E., & Chugunov, A. I. (2014). Quasinormal modes of superfluid neutron stars. Physical Review D, 90, 024010.  https://doi.org/10.1103/PhysRevD.90.024010, arXiv:1404.7512.
  25. Karami, K. (2008). Third order effect of rotation on stellar oscillations of a B star. Chinese Journal of Astronomy and Astrophysics, 8, 285–308.  https://doi.org/10.1088/1009-9271/8/3/06, ArXiv: astro-ph/0502194.ADSCrossRefGoogle Scholar
  26. Karami, K. (2009). Third order effect of rotation on stellar oscillations of a \(\beta \)-Cephei star. Astrophysics and Space Science, 319, 37–44.  https://doi.org/10.1007/s10509-008-9937-x, arXiv:0810.5092.ADSMathSciNetCrossRefGoogle Scholar
  27. Kojima, Y. (1992). Equations governing the nonradial oscillations of a slowly rotating relativistic star. Physical Review D, 46, 4289–4303.  https://doi.org/10.1103/PhysRevD.46.4289.ADSMathSciNetCrossRefGoogle Scholar
  28. Kojima, Y. (1993). Coupled pulsations between polar and axial modes in a slowly rotating relativistic star. Progress of Theoretical Physics, 90, 977–990.  https://doi.org/10.1143/PTP.90.977.ADSCrossRefGoogle Scholar
  29. Kokkotas, K. D., & Schmidt, B. (1999). Quasi-normal modes of stars and black holes. Living Reviews in Relativity, 2, 2.  https://doi.org/10.12942/lrr-1999-2, arXiv:gr-qc/9909058.
  30. Kokkotas, K. D., & Schutz, B. F. (1986). Normal modes of a model radiating system. General Relativity and Gravitation, 18, 913–921.  https://doi.org/10.1007/BF00773556.ADSCrossRefGoogle Scholar
  31. Kokkotas, K. D., & Schutz, B. F. (1992). W-modes: A new family of normal modes of pulsating relativistic stars. Monthly Notices of the Royal Astronomical Society, 255, 119–128.  https://doi.org/10.1093/mnras/255.1.119.ADSCrossRefGoogle Scholar
  32. Kokkotas, K. D., & Stergioulas, N. (1999). Analytic description of the r-mode instability in uniform density stars. Astronomy & Astrophysics, 341, 110–116. http://adsabs.harvard.edu/abs/1999A%26A...341.110K, arXiv: astro-ph/9805297.
  33. Krüger, C. J., Ho, W. C. G., & Andersson, N. (2015). Seismology of adolescent neutron stars: Accounting for thermal effects and crust elasticity. Physical Review D, 92, 063009.  https://doi.org/10.1103/PhysRevD.92.063009, arXiv:1402.5656.
  34. Ledoux, P., & Walraven, T. (1958). Variable stars. Encyclopedia of physics (Handbuch der Physik) (vol. 51, pp. 353–604). Berlin-Göttingen-Heidelberg: Springer-Verlag. http://adsabs.harvard.edu/abs/1958HDP....51..353L.CrossRefGoogle Scholar
  35. Lignières, F., Rieutord, M., & Reese, D. (2006). Acoustic oscillations of rapidly rotating polytropic stars. I. Effects of the centrifugal distortion. Astronomy & Astrophysics, 455, 607–620.  https://doi.org/10.1051/0004-6361:20065015, arXiv:astro-ph/0604312.ADSCrossRefGoogle Scholar
  36. Lindblom, L., & Detweiler, S. L. (1983). The quadrupole oscillations of neutron stars. The Astrophysical Journal Supplement Series, 53, 73–92.  https://doi.org/10.1086/190884.ADSCrossRefGoogle Scholar
  37. Lindblom, L., & Ipser, J. R. (1999). Generalized r-modes of the Maclaurin spheroids. Physical Review D, 59, 044009.  https://doi.org/10.1103/PhysRevD.59.044009, arXiv:gr-qc/9807049.
  38. Lockitch, K. H., & Friedman, J. L. (1999). Where are the r-modes of isentropic stars? The Astrophysical Journal, 521, 764–788.  https://doi.org/10.1086/307580, arXiv:gr-qc/9812019.ADSCrossRefGoogle Scholar
  39. Lynden-Bell, D., & Ostriker, J. P. (1967). On the stability of differentially rotating bodies. Monthly Notices of the Royal Astronomical Society, 136, 293. http://adsabs.harvard.edu/abs/1967MNRAS.136.293L.ADSCrossRefGoogle Scholar
  40. McDermott, P. N. (1990). Density discontinuity g-modes. Monthly Notices of the Royal Astronomical Society, 245, 508. http://adsabs.harvard.edu/abs/1990MNRAS.245.508M.
  41. McDermott, P. N., van Horn, H. M., & Scholl, J. F. (1983). Nonradial g-mode oscillations of warm neutron stars. The Astrophysical Journal, 268, 837–848. http://dx.doi.org/10.1086/161006.ADSCrossRefGoogle Scholar
  42. McDermott, P. N., van Horn, H. M., & Hansen, C. J. (1988). Nonradial oscillations of neutron stars. The Astrophysical Journal, 325, 725–748.  https://doi.org/10.1086/166044.ADSCrossRefGoogle Scholar
  43. Miniutti, G., Pons, J. A., Berti, E., Gualtieri, L., & Ferrari, V. (2003). Non-radial oscillation modes as a probe of density discontinuities in neutron stars. Monthly Notices of the Royal Astronomical Society, 338, 389–400.  https://doi.org/10.1046/j.1365-8711.2003.06057.x, arXiv:astro-ph/0206142.ADSCrossRefGoogle Scholar
  44. Osaki, J. (1975). Nonradial oscillations of a 10 solar mass star in the main-sequence stage. Publications of the Astronomical Society of Japan, 27, 237–258. http://adsabs.harvard.edu/abs/1975PASJ...27.237O.
  45. Papaloizou, J., & Pringle, J. E. (1978). Non-radial oscillations of rotating stars and their relevance to the short-period oscillations of cataclysmic variables. Monthly Notices of the Royal Astronomical Society, 182, 423–442. http://adsabs.harvard.edu/abs/1978MNRAS.182.423P.ADSCrossRefGoogle Scholar
  46. Pekeris, C. L. (1938). Nonradial oscillations of stars. The Astrophysical Journal, 88, 189.  https://doi.org/10.1086/143971.ADSCrossRefGoogle Scholar
  47. Reese, D. R. (2010). Oscillations in rapidly rotating stars. Astronomische Nachrichten, 331, 1038.  https://doi.org/10.1002/asna.201011452.ADSCrossRefGoogle Scholar
  48. Reese, D., Lignières, F., & Rieutord, M. (2006). Acoustic oscillations of rapidly rotating polytropic stars. II. Effects of the Coriolis and centrifugal accelerations. Astronomy & Astrophysics, 455, 621–637. arXiv:astro-ph/0605503.ADSCrossRefGoogle Scholar
  49. Reisenegger, A., & Goldreich, P. (1992). A new class of g-modes in neutron stars. The Astrophysical Journal, 395, 240–249.  https://doi.org/10.1086/171645.ADSCrossRefGoogle Scholar
  50. Robe, H. (1968). Les oscillations non radiales des polytropes. Annales d’Astrophysique, 31, 475. http://adsabs.harvard.edu/abs/1968AnAp...31..475R.
  51. Saio, H. (1981). Rotational and tidal perturbations of nonradial oscillations in a polytropic star. The Astrophysical Journal, 244, 299–315.  https://doi.org/10.1086/158708.ADSCrossRefGoogle Scholar
  52. Saio, H. (1982). R-mode oscillations in uniformly rotating stars. The Astrophysical Journal, 256, 717–735.  https://doi.org/10.1086/159945.ADSMathSciNetCrossRefGoogle Scholar
  53. Sauvenier-Goffin, E. (1951). Note sur les pulsations non-radiale d’une sphère homogène compressible. Bulletin de la Societe Royale des Sciences de Liège, 20, 20–38. http://adsabs.harvard.edu/abs/1951BSRSL.20...20S.
  54. Schenk, A. K., Arras, P., Flanagan, É. É., Teukolsky, S. A., & Wasserman, I. (2001). Nonlinear mode coupling in rotating stars and the r-mode instability in neutron stars. Physical Review D, 65, 024001.  https://doi.org/10.1103/PhysRevD.65.024001, arXiv:gr-qc/0101092.
  55. Smeyers, P. (2003). Asymptotic representation of low- and intermediate-degree p-modes in stars. Astronomy & Astrophysics, 407, 643–653.  https://doi.org/10.1051/0004-6361:20030744.ADSMathSciNetCrossRefGoogle Scholar
  56. Smeyers, P. (2006). The second-order asymptotic representation of higher-order non-radial p-modes in stars revisited. Astronomy & Astrophysics. 451, 237–249.  https://doi.org/10.1051/0004-6361:20054546.ADSCrossRefGoogle Scholar
  57. Smeyers, P., & van Hoolst, T. (2010). Linear isentropic oscillations of stars: Theoretical foundations (vol. 371). Astrophysics and Space Science Library. New York: Springer. http://adsabs.harvard.edu/abs/2010ASSL..371.....S.CrossRefGoogle Scholar
  58. Smeyers, P., De Boeck, I., Van Hoolst, T., & Decock, L. (1995). Asymptotic representation of linear, isentropic g-modes of stars. Astronomy & Astrophysics, 301, 105. http://adsabs.harvard.edu/abs/1995A%26A...301.105S.
  59. Smeyers, P., Vansimpsen, T., De Boeck, I., & Van Hoolst, T. (1996). Asymptotic representation of high-frequency, low-degree p-modes in stars and in the Sun. Astronomy & Astrophysics, 307, 105. http://adsabs.harvard.edu/abs/1996A%26A...307.105S.
  60. Soufi, F., Goupil, M. J., & Dziembowski, W. A. (1998). Effects of moderate rotation on stellar pulsation. I. Third order perturbation formalism. Astronomy & Astrophysics, 334, 911–924. http://adsabs.harvard.edu/abs/1998A%26A...334.911S.
  61. Stavridis, A., & Kokkotas, K. D. (2005). Evolution equations for slowly rotating stars. International Journal of Modern Physics D, 14, 543–571.  https://doi.org/10.1142/S021827180500592X, arXiv:gr-qc/0411019.ADSMathSciNetCrossRefGoogle Scholar
  62. Stergioulas, N. (2003). Rotating stars in relativity. Living Reviews in Relativity, 6, 3.  https://doi.org/10.12942/lrr-2003-3, arXiv:gr-qc/0302034.
  63. Strohmayer, T. E. (1991). Oscillations of rotating neutron stars. The Astrophysical Journal, 372, 573–591.  https://doi.org/10.1086/170002.ADSCrossRefGoogle Scholar
  64. Strohmayer, T. E. (1993). Density discontinuities and the g-mode oscillation spectra of neutron stars. The Astrophysical Journal, 417, 273.  https://doi.org/10.1086/173309.ADSCrossRefGoogle Scholar
  65. Thomson, W. (1863). Dynamical problems regarding elastic spheroidal shells and spheroids of incompressible liquid. Philosophical Transactions of the Royal Society of London, 153, 583616.  https://doi.org/10.1098/rstl.1863.0028.ADSCrossRefGoogle Scholar
  66. Thorne, K. S., & Campolattaro, A. (1967). Non-radial pulsation of general-relativistic stellar models. I. Analytic analysis for \(l\ge 2\). The Astrophysical Journal, 149, 591.  https://doi.org/10.1086/149288.ADSCrossRefGoogle Scholar
  67. Unno, W., Osaki, Y., Ando, H., Saio, H., & Shibahashi, H. (1989). Nonradial oscillations of stars (2nd ed.). Tokyo: University of Tokyo Press. http://adsabs.harvard.edu/abs/1989nos..book.....U.
  68. Vavoulidis, M., Stavridis, A., Kokkotas, K. D., & Beyer, H. (2007). Torsional oscillations of slowly rotating relativistic stars. Monthly Notices of the Royal Astronomical Society, 377, 1553–1556.  https://doi.org/10.1111/j.1365-2966.2007.11706.x, arXiv:gr-qc/0703039.ADSCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Mathematical Sciences and STAG Research CentreUniversity of SouthamptonSouthamptonUK

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