IJG  Vol.5 No.6 , May 2014
The Physics of Rotational Flattening and the Point Core Model
Abstract: The effect of rotation on the shape (figure) and gravitational quadrupole of astronomical bodies is calculated by using an approximate point core model: A point mass at the center of an ellipsoidal homogeneous fluid. Maclaurin’s analytical result for homogenous bodies generalizes to this model and leads to very accurate analytical results connecting the three observables: oblateness (ò), gravitational quadrupole (J2), and angular velocity parameter (q). The analytical results are compared to observational data for the planets and a good agreement is found. Oscillations near equilibrium are studied within the model.
Cite this paper: Essén, H. (2014) The Physics of Rotational Flattening and the Point Core Model. International Journal of Geosciences, 5, 555-570. doi: 10.4236/ijg.2014.56051.

[1]   Todhunter, I. (1873) A History of the Mathematical Theories of Attraction and the Figure of the Earth. Macmillan and Company, London. (Reprinted: Dover Publications, New York, 1962)

[2]   Will, C.M. (1993) Theory and Experiment in Gravitational Physics. Cambridge University Press, Cambridge.

[3]   Godier, S. and Rozelot, J.-P. (1999) Quadrupole Moment of the Sun. Gravitational and Rotational Potentials. Astronomy & Astrophysics, 350, 310-317.

[4]   Laskar, J. (1999) The Limits of Earth Orbital Calculations for Geological Timescale Use. Philosophical Transactions of the Royal Society A, 357, 1735-1759.

[5]   Jeffreys, H. (1962) The Earth, Its Origin History and Physical Constitution. 4th Edition, Cambridge University Press, Cambridge.

[6]   Jardetzky, W.S. (1958) Theories of Figures of Celestial Bodies. Interscience, New York.

[7]   Zharkov, V.N. and Trubitsyn, V.P. (Editor Hubbard, W.B.) (1978) Physics of Planetary Interiors. Pachart Publishing House, Tuscon.

[8]   Cook, A.H. (1980) Interiors of the Planets. Cambridge University Press, Cambridge.

[9]   Moritz, H. (1990) The Figure of the Earth. Wichmann, Karlsruhe.

[10]   Chandrasekhar, S. (1969) Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven and London.

[11]   Murray, C.D. and Dermott, S.F. (1999) Solar System Dynamics. Cambridge University Press, Cambridge.

[12]   Kaula, W.M. (2000) Theory of Satellite Geodesy. Dover, Mineola.

[13]   Kippenhahn, R. and Weigert, A. (1990) Stellar Structure and Evolution. Springer-Verlag, Berlin.

[14]   Hubbard, W.B. and Anderson, J.D. (1978) Possible Flyby Measurements of Galilean Satellite Interior Structure. Icarus, 33, 336-341.

[15]   Dermott, S.F. and Thomas, P.C. (1988) The Shape and Internal Structure of Mimas. Icarus, 73, 25-65.

[16]   Abad, S., Pacheco, A.F. and Sanudo, J. (1995) Variational Methods to Calculate the Hydrostatic Structure of Rotating Planets. Geophysical Journal International, 122, 953-960.

[17]   Denis, C., Tomecka-Suchon, S., Rogister, Y. and Amalvict, M. (2000) Comment on “Variational Methods to Calculate the Hydrostatic Structure of Rotating Planets” by S. Abad, A. F. Pacheco, and J. Sa-udo. Geophysical Journal International, 143, 985-986.

[18]   Denis, C., Amalvict, M., Rogister, Y. and Tomecka-Suchon, S. (1998) Methods for Computing Internal Flattening, with Applications to the Earth’s Structure and Geodynamics. Geophysical Journal International, 132, 603-642.

[19]   Stacey, F.D. (1969) Physics of the Earth. John Wiley, New York.

[20]   MacMillan, W.D. (1958) The Theory of the Potential. Dover, New York.

[21]   Landau, L. and Lifshitz, E.M. (1975) The Classical Theory of Fields. 4th Edition, Pergamon, Oxford.

[22]   Dankova, Ts. and Rosensteel, G. (1998) Triaxial Bifurcations of Rapidly Rotating Spheroids. American Journal of Physics, 66, 1095-1100.

[23]   Lodders, K. and Fegley Jr., B. (1998) The Planetary Scientist’s Companion. Oxford University Press, New York.

[24]   Anderson, J.D. and Schubert, G. (2007) Saturn’s Gravitational Field, Internal Rotation, and Interior Structure. Science, 317, 1384-1387.

[25]   Essen, H. (1993) Average Angular Velocity. European Journal of Physics, 14, 201-205.

[26]   Ulrich, R.K. and Hawkins, G.W. (1981) The Solar Gravitational Figure—J2 and J4. The Astrophysical Journal, 246, 985-988.

[27]   Paterno, L., Sofia, S. and Di Mauro, M.P. (1996) The Rotation of the Sun’s Core. Astronomy & Astrophysics, 314, 940-946.

[28]   Lydon, T.J. and Sofia, S. (1996) A Measurement of the Shape of the Solar Disc (the Solar Quadrupole Moment, the Solar Octopole Moment, and the Advance of the Perihelion of the Planet Mercury). Physical Review Letters, 76, 177179.

[29]   Udias, A. (1999) Principles of Seismology. Cambridge University Press, Cambridge.

[30]   Essen, H. (1996) A Simple Mechanical Model for the Shape of the Earth. European Journal of Physics, 17, 131-135.