DDA 36th Meeting, 10-14 April 2005
Session 7 Planets: Orbits and Tides
Oral, Tuesday, April 12, 2005, 9:35am-12:15pm

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[7.06] Singularities in the Elastic Tidal Deformation of a Sphere with Radially Varying Density and Elasticity

S.E. Frey (California State University, Hayward), T. Hurford, R. Greenberg (University of Arizona)

Love’s classical (1911) linear elastic model for the tidal amplitude of a self-gravitating body assumes that prior to application of the external tidal potential the body is uniform in density and elastic parameters. Numerical evaluations of the solution to Love's governing equations reveal portions of parameter space for which infinitesimal tide raisers can raise tides of arbitrary height (Hurford et. al. this conference). However, a homogeneously dense sphere is a rather non-physical initial condition for tidal deformations. Here we consider a more physically relevant model, that of a continuous density profile which varies with radius. We also allow for the radially varying elastic parameters. We show that the singularities observed in the homogeneous case persist.

We assume the density, rigidity and/or Lamè constant are of the form of a polynomial in the normalized radius. The solution to the equations governing the tidal deformation of a radially varying sphere depends only on the effective gravitational rigidity, \rho g R / \mu, and the ratio of rigidity to Lamé constant, \mu / \lambda . As the magnitude of the radial density variation is increased, the locations of the singularities are displaced in parameter space. Through study of the change in singularity location as the profiles of density, rigidity and compressibility are varied, we can investigate the forces dominating the occurrence of the singular type behavior: Self-gravitation is the force driving the large amplitude tidal deformations.

To elucidate the effect further, we developed a non-linear elastic formulation of the self-gravitational collapse of a homogeneous sphere, which reveals material parameter values for which the sphere is unstable. The singularities that we mapped represent the effects of these instabilities in self-gravitation.

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Bulletin of the American Astronomical Society, 37 #2
© 2005. The American Astronomical Soceity.