The aim of the present paper will be to formulate explicitly the differential equations which govern three-dimensional rotation of deformable self-gravitating bodies of arbitrary structure, and consisting of fluid material whose viscosity is an arbitrary function of spatial coordinates; with special respect to a description of the effects of viscous friction exhibited in binary systems which consist of a close pair of such configurations.
In Section 2 which follows brief introductory remarks outlining the problem, we depart from the fundamental equations expressed in terms of the velocity components of deformation, arising from dynamical tides that are raised on the rotating configuration by the attraction of its companion. In Section 3, the magnitude of such velocities in the rotating frame of reference (arising from possible eccentricity of the orbit of the two bodies, as well as from the asynchronism between their rotation and revolution) will be specified in terms of the physical characteristics of each particular system; and in Section 4 we shall similarly evaluate the moments and products of inertia of the rotating configuration.
In Section 5, we shall focus our attention to the effects, on rotational motion, of the time-dependent deformations which the components undergo in the course of each orbital cycle as a result of the dynamical tides; and in Section 6 we shall detail the general effects on rotation arising from the viscosity; while in the subsequent Section 7 we shall specify these more explicitly for the case of simple rotation about an axis perpendicular to the orbital plane of the system.
A numerical application of the results of this section to the Earth-Moon system (the components of which will be regarded as yielding to mutual tidal distortion) discloses that-at the present distance of these bodies-the rate at which viscous friction of bodily tides tends to equalize their rotation and revolution works some 104 times more effectively on the Moon than on the Earth. This should satisfactorily account for the fact that the Moon has attained a state of synchronous rotation (and probably maintained it at most times throughout its long astronomical past), while our Earth is still far from this state today.
In the concluding Section 8 of this paper we shall generalize the results of Section 7 to the case of spheroidal configurations, the axial rotation of which is fast enough for the cross-products between rotational and tidal distortion to become significant. Lastly, in Appendix A we shall express the velocity components arising from dynamical tides in terms of the spherical polar (rather than rectangular) corrdinates; and evaluate the rate of the dissipation of the kinetic energy into heat, through viscous tides, of different type, within the orbital cycle (Appendix B).
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Banachiewicz, Th.: 1950,Trans. IAU,7, 174.
Chapman, S.: 1954,Astrophys. J. 120, 151.
Darwin, G. H.: 1910,Scientific Papers 2, 36–139 (published originally inPhil. Trans. Roy. Soc. London 170, 447, 1879).
Hazlehurst, J. and Sargent, W. L. W.: 1959,Astrophys. J. 130, 276.
Jeffreys, H.: 1970,The Earth (5th ed.), Cambr. Univ. Press, pp. 327 or 423.
Kopal, Z.: 1953,Monthly Notices. Roy. Astron. Soc. 113, 769.
Kopal, Z.: 1959,Close Binary Systems, Chapman-Hall and John Wiley, London and New York, Section II. 2.
Kopal, Z.: 1968,Astrophys. Space Sci. 1, 74 (‘Paper I’).
Kopal, Z.: 1969a,Astrophys. Space Sci. 4, 320, (‘Paper II’).
Kopal, Z.: 1969b,Astrophys. Space Sci. 4, 427 (‘Paper III’).
Kopal, Z.: 1969c,The Moon (2nd ed.), D. Reidel Publ. Co., Dordrecht-Holland, pp. 203–204.
Lamb, H.: 1932,Hydrodynamics, Cambridge Univ. Press, 6th ed., p. 580.
Urey, H. C.: 1968,Science 162, 1408.
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Kopal, Z. The effects of viscous friction on axial rotation of celestial bodies. Astrophys Space Sci 16, 3–51 (1972). https://doi.org/10.1007/BF00643090
- Axial Rotation
- Celestial Body
- Viscous Friction
- Introductory Remark
- Close Pair