Abstract
The aim of the present investigation has been to establish the minimum distance (commonly referred to as the ‘Roche limit’), to which a small satellite can approach its central star without the loss of its stability.
In order to do so, we shall depart from hydrodynamical equations governing small oscillations of stellar structures, and set out to establish the limit at which their distorted form of equilibrium can no longer vibrate periodically in response to arbitrary perturbations. To this end, such equations will be rewritten in terms of curvilinear Clairaut coordinates (Kopal, 1980) in which the gravitational potential defining equilibrium surfaces plays the role of the radial coordinate; and their solution constructed for the classical Roche problem in which the oscillating satellite of infinitesimal mass consists of material which is homogeneous and incompressible, while its primary component acts gravitationally as a mass-point.
The outcome of such a solution agrees satisfactorily with that previously established by Chandrasekhar (1963) on the basis of the virial theorem; but the method employed by us lends itself more readily to a generalization of the Roche limit to systems of finite mass ratios and consisting of the components of finite size.
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References
Bryan, G. H.: 1888,Phil. Trans. Roy. Soc. London A180, 187.
Chandrasekhar, S.: 1963,Astrophys. J. 138, 1182.
Darwin, G. H.: 1906,Phil. Trans. Roy. Soc. London A206, 161.
Darwin, G. H.: 1911,The Tides, 3rd ed., John Murray, London, p. 436.
Jeans, J. H.: 1919,Problems of Cosmogony and Stellar Dynamics, Cambridge University Press, pp. 52–53.
Kopal, Z.: 1960,Figures of Equilibrium of Celestial Bodies, Wisconsin University Press.
Kopal, Z.: 1978,Dynamics of Close Binary Systems, D. Reidel Publ. Co., Dordrecht, Holland.
Kopal, Z.: 1980,Astrophys. Space Sci. 70, 407.
Kopal, Z.: 1981,Astrophys. Space Sci. 76, 187.
Kopal, Z.: 1983,Astrophys. Space Sci. 93, 149.
Kopal, Z. and Song, G.-X.: 1983,Astrophys. Space Sci. 92, 3.
Roche, E. A.: 1850,Mém. de l'Acad. des Sci. de Montpellier 1, 243.
Struve, O.: 1961,The Universe, M.I.T. Press, Cambridge, Mass., p. 18.
Thomson, W.: 1863,Phil. Trans. Roy. Soc. London A153, 583 (cf. pp. 608ff).
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Kopal, Z., Song, GX. Roche limit for homogeneous incompressible masses. Astrophys Space Sci 96, 381–404 (1983). https://doi.org/10.1007/BF00651683
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DOI: https://doi.org/10.1007/BF00651683