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Dynamical tides in close binary systems

III: Dissipation of energy

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Abstract

The aim of the present paper will be to investigate the circumstances under which an irreversible dissipation of the kinetic energy into heat is generated by the dynamical tides in close binary systems if (a) their orbit is eccentric; (b) the axial rotation of the components is not synchronized with the revolution; or (c) the equatorial planes are inclined to that of the orbit.

In Section 2 the explicit form of the viscous dissipation function will be set up in terms of the velocity-components of spheroidal deformation arising from the tides; in Section 3, the principal partial tides contributing to the dissipation will be detailed; Section 4 will be devoted to a determination of the extent of stellar viscosity — both gas and radiative; while in the concluding Section 5 quantitative estimates will be given of the actual rate at which the kinetic energy of dynamical tides gets dissipated into heat by viscous friction in stellar plasma.

The results disclose that the amount of heat produced per unit time by tidal interaction between components of actual close binaries equals only about 10−10th part of their nuclear energy production; and cannot, therefore, affect the internal structure of evolution of the constituent stars to any appreciable extent. Moreover, it is shown that the kinetic energy of their axial rotation can be influenced by tidal friction only on a nuclear, rather than gravitational (Kelvin) time-scale — as long as plasma or radiative viscosity constitute the sole sources of dissipation. However, the emergence of turbulent viscosity in secondary components of late spectral types, which have evolved away from the Main Sequence, can accelerate the dissipation 105–106 times, and thus give rise to appreciable changes in the elements of the system (particularly, in the orbital periods) over time intervals of the order of 105–106 years. Lastly, it is pointed out that, in close binary systems consisting of a pair of white dwarfs, a dissipation of the kinetic energy through viscous tides in degenerate fermion-gas could produce enough heat to account, by itself, for the observed luminosity of such objects.

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References

  1. Chapman, S.: 1954, ‘The Viscosity and Thermal Conductivity of a Completely Ionized Gas’,Astrophys. J. 120, 151–155.

  2. Eddington, A.S.: 1926,The Internal Constitution of the Stars. Cambridge University Press, Cambridge, sec. 197.

  3. Hazlehurst, J. andSargent, W.L.W.: 1959, ‘Hydrodynamics in a Radiation Field — a Covariant Treatment’,Astrophys. J. 130, 276–285.

  4. Kopal, Z.: 1957, Presidential Report of Commission 42,Trans. I.A.U.,9, pp. 611–612.

  5. Kopal, Z.: 1968a, ‘Dynamical Tides in Close Binary Systems. I’,Astrophys. Space Sci.,1, 179.

  6. Kopal, Z.: 1968b, ‘Dynamical Tides in Close Binary Systems. II’,Astrophys. Space Sci.,1, 284.

  7. Kopal, Z. andShapley, M.B.: 1956, ‘Catalogue of the Elements of Eclipsing Binary Systems’,Jodrell Bank Annals 1, 141–221.

  8. Lamb, H.: 1932,Hydrodynamics (6th ed). Cambridge University Press, Cambridge, sec 329.

  9. Motz, L.: 1952, ‘On the Radius of Gyration of Stars’,Astrophys. J. 115, 562–566.

  10. Nishimura, H. andMori, H.: 1961, ‘Viscosity of Fermi-Particle Systems’, inProgress in Theoretical Physics 26, 967–989.

  11. Oster, L.: 1957, ‘Viskosität, elektrische und thermische Leitfähigkeit stellarer Materie’,Z. Astrophys. 42, 228–262.

  12. Schwarzschild, M.: 1958,Structure and Evolution of the Stars. Princeton University Press, Princeton.

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Kopal, Z. Dynamical tides in close binary systems. Astrophys Space Sci 1, 411–423 (1968). https://doi.org/10.1007/BF00658765

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Keywords

  • Axial Rotation
  • Viscous Dissipation
  • Spectral Type
  • Dissipation Function
  • Secondary Component