**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.