Generalized r-modes of the Maclaurin spheroids

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 59, Issue 4, 1999, Pages 1-13

  Lindblom, Lee ^a   Ipser, James R ^b  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b University of Florida   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17144395959     PISSN: 05562821     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevD.59.044009     Document Type: Article

References (16)

1. N. Andersson, Astrophys. J. 502, 708 (1998).
2. J. L. Friedman, and S. M. Morsink, Astrophys. J. 502, 714 (1998).
3. L. Lindblom, B. J. Owen, and S. M. Morsink, Phys. Rev. Lett. 80, 4843 (1998).
4. B. J. Owen, L. Lindblom, C. Cutler, B. F. Schutz, A. Vecchio, and N. Andersson, Phys. Rev. D 58, 084020 (1998).
5. G. H. Bryan, Philos. Trans. R. Soc. London A180, 187 (1889).
6. lm [J. Papaloizou and J. E. Pringle, Mon. Not. R. Astron. Soc. 182, 423 (1978)]. Such modes were called r-modes because they are analogous to the Rossby waves of atmospheric physics. The modes studied here do not fit neatly into the traditional classification of the r- and g-modes of slowly rotating stars. The velocity perturbations of our modes are not in general purely axial (as are the traditional r-modes) nor purely polar (as are the traditional g-modes) in the small angular velocity limit. Thus we have extended the traditional definition of r-mode by defining it in terms of the physical process that governs the mode, rotation, rather than the symmetry of its velocity perturbation. Our generalized r-modes include all modes considered r-modes under the traditional definition.
7. The Maclaurin spheroids are a special case of barotropic stellar models: stars in which the density of the perturbed fluid depends only on the pressure. In non-barotropic stars there also exist buoyancy forces that dominate the behavior of a class of modes called g-modes. The non-barotropic analogues of the generalized r-modes discussed here will almost certainly be influenced at some level by buoyancy forces. At sufficiently small angular velocities (i.e. at angular velocities smaller than the Brunt-Väisälä frequency) buoyancy forces could well dominate the dynamics of some of these modes; and such modes might then be called generalized g-modes in non-barotropic models. By analogy some might prefer to call these modes generalized g-modes even in the barotropic case. Since we do not at present know which (if any) of these modes might be dominated by buoyancy forces in the non-barotropic case, we prefer to refer to them here as r-modes. In the barotropic case the dynamics of these modes are dominated by rotational forces.
8. J. Papaloizou and J. E. Pringle, Mon. Not. R. Astron. Soc. 182, 423 (1978).
9. It has been traditional in the literature to discuss the r-modes in terms of a vector-spherical-harmonic decomposition of the Lagrangian displacement. In terms of that traditional decomposition, the "classical" r-modes correspond to the case l =m. The analysis presented here uses a certain scalar potential δU from which the Lagrangian displacement is determined. The analytic solutions that we find for δU are particular spheroidal harmonic functions. The classical r-modes correspond to the case l=m+1 in terms of these spheroidal harmonics. The potential δU for the classical r-modes can also be expressed as a spherical harmonic with l=m+1. In general, however, the modes discussed here do not have simple finite representations in terms of spherical harmonics.
10. J. R. Ipser and L. Lindblom, Astrophys. J. 355, 226 (1990).
11. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale University Press, New Haven, 1969).
12. H. Bateman, in Higher Transcendental Functions, edited by A. Erdélyi (McGraw-Hill, New York, 1953), Vol. I.
13. 0<2. These are all of the zeros of the right side of Eq. (6.5) because it is a polynomial of degree l-m.
14. 3 order. Kokkotas and Stergioulas only claim that their expression is an approximation.
15. 3 order. Kokkotas and Stergioulas only claim that their expression is an approximation.
16. J. Provost, G. Berthomieu, and A. Rocca, Astron. Astrophys. 94, 126 (1981).