Abstract
According to the classical theory of equilibrium figures, surfaces of equal density, potential and pressure concur (let us call them isobars). Isobars can be represented by means of Liapunov power series in small parameter q, up to the first approximation coinciding with the centrifugal to gravitational force ratio at the equator. Liapunov has proved the existence of the universal convergence domain: the above mentioned series converge for all bodies (satisfying a natural condition that the density ρ decreases from the center to the surface) if |q| < q*. Using Liapunov’s algorithm and symbolic manipulation tools, we have found q*= 0.000370916. Evidently, the convergence radius q* may be much greater in common situations. To comfirm it it is reasonable to consider two limiting and one or two intermediate cases for the density behaviour: ρ is a constant, the Dirac’s δ-function, linear function of the distance from the center, etc. And indeed, in the previous paper we find a three orders of magnitude greater value for homogeneous figures. In the present paper we find that in the opposite case of Huygens-Roche figures (a point-mass surrounded by a weightless atmosphere) the convergence radius is unexpectedly large and coincides with the well-known biggest possible value q*= 0.541115598 for such a class of figures. To ascertain it we ought to use numerical calculations, so our main result is demonstrated by means of a computer assisted proof.
Similar content being viewed by others
References
von Lichtenstein, L.: 1933, Gleichgewichtsfiguren rotierender Flüssigkeiten, Springer-Verlag, Berlin.
Appell, P.: 1932, Traité de Mécanique Rationnelle, T. 4, Fasc. 1, Gauthier-Villars, Paris.
Liapounoff, A.M.: 1903, ‘Recherches dans la théorie de la figure des corps célestes', Notes de l'Académie Impériale des Sci. 14(7), 1-37.
Zaky, S.F., Elkin, A.V. and Kholshevnikov, K.V.: 1994, ‘Form and structure of celestial bodies: in the wake of Liapunov', Astron. Zh. 71(5), 785-793 (in Russian). English version: 1994, Astron. Rep. 38(5), 699-707.
Kholshevnikov, K.V. and Elkin, A.V.: 2002, ‘Convergence of Liapunov Series for Maclaurin ellipsoids', Celest. Mech. Dyn. Astr. 84(1), 57-63.
Kholshevnikov, K.V. and Kurdubov, S.L.: 2002, ‘On convergence radius of Liapunov series for Huygens-Roche figures', Vestnik St. Petersburg Univ. Ser. 1(1), 81-92 (in Russian).
Korn, G. and Korn, T.: 1968, Mathematical Handbook for Scientists and Engineers. Second, Enlarged and Revised Edition, McGraw-Hill Book Co., New York, Toronto, Ont., London.
Duren, P.: 1970, Theory of Hp Spaces, Academic Press, New York and London.
Kholshevnikov, K.V.: 2003, ‘Convergence of Liapunov series for Maclaurin ellipsoids: real analysis', Celest. Mech. Dyn. Astr. 87(?), ??-??.
Bateman, H., Erdélyi, A.: 1953, Higher Transcendental Functions, McGraw-Hill Book Co., New York, Toronto, London.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kholshevnikov, K.V., Kurdubov, S.L. Convergence of Liapunov series for Huygens–Roche Figures. Celestial Mechanics and Dynamical Astronomy 89, 83–96 (2004). https://doi.org/10.1023/B:CELE.0000028169.90756.17
Issue Date:
DOI: https://doi.org/10.1023/B:CELE.0000028169.90756.17