Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Total gravitational energy of a slightly ellipsoidal trilayer planet


Authors: Pavel Grinfeld and Jack Wisdom
Journal: Quart. Appl. Math. 64 (2006), 271-281
MSC (2000): Primary 81V17
DOI: https://doi.org/10.1090/S0033-569X-06-00985-5
Published electronically: April 17, 2006
MathSciNet review: 2243863
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Abstract | References | Similar Articles | Additional Information

Abstract: We use a perturbational technique to compute the total gravitational energy of a slightly ellipsoidal trilayer planet as a function of the two sets of Euler angles. A second order computation is required since the torque is proportional to the product of the ellipticities of the inner core and the mantle. Although we focus on ellipsoidal perturbations, the intermediate analytical expressions are valid for arbitrary small deformations of the spherical configuration.

The primary application of the expression for the total gravitational energy is in the Lagrangian formulation of dynamics. As a by-product, we determine the gravitationally stable equilibrium orientation of the rigid inner core. Due to symmetry, the six coaxial configurations are equilibrium. We show how to identify the stable configuration and to prove its uniqueness.


References [Enhancements On Off] (What's this?)

  • 1. Grinfeld P., 2003. Boundary Perturbations of Laplace Eigenvalues. Applications to Polygons and Electron Bubbles. Thesis, Department of Mathematics, MIT.
  • 2. Grinfeld P., Wisdom J. (Submitted). A way to compute the gravitational potential for near-spherical geometries.
  • 3. G. Schubert, M.N. Ross, D.J. Stevenson, and T. Spohn, 1988. Mercury's Thermal History and the Generation of Its Magnetic Field. Mercury (Univ. of Ariz Press), 429-460.
  • 4. A.M.K. Szeto and S. Xu (1997). Gravitational Coupling in a Triaxial Ellipsoidal Earth, J. Geophys. Res., vol. 102, 27651-27657.
  • 5. Gerald Jay Sussman, Jack Wisdom, and Meinhard E. Mayer, Structure and interpretation of classical mechanics, MIT Press, Cambridge, MA, 2001. MR 1825485

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Additional Information

Pavel Grinfeld
Affiliation: Department of Mathematics, Drexel University

Jack Wisdom
Affiliation: Department of EAPS, MIT

DOI: https://doi.org/10.1090/S0033-569X-06-00985-5
Received by editor(s): May 10, 2005
Published electronically: April 17, 2006
Article copyright: © Copyright 2006 Brown University


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