**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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## [7.05] Singularities in the Elastic Tidal Deformation of a Homogeneous Sphere

*T.A. Hurford (University of Arizona), S.E. Frey (California State University - East Bay), R. Greenberg (University of Arizona)*

Numerical evaluation of Love's 1911 solution [1] for the
tidal amplitude of a uniform, compressible, self-gravitating
body reveals portions of parameter space where the
coefficients known as Love numbers approach infinity. Love's
solution depends only on (a) the ratio of gravity to the
rigidity, \rho g R / \mu , and (b) the ratio of rigidity
to Lamé constant, \mu / \lambda . The solution is not
continuous; it includes the above singularities, around
which values approach plus-or-minus infinity, even for
parameters appropriate for realistic planetary materials.
However, Love’s model assumed that initially (prior to
imposing the tidal potential) the sphere is homogeneous in
density throughout the body. This condition is artificial,
because for a real body, self-gravity would have increased
the internal density. Nevertheless, Love’s solutions have
been the standard textbook formula for tidal amplitudes,
usually only considering the incompressible limit. The
singularities occur only for specific values of the
parameters when compressibility is taken into account. For
most typical planetary materials, with compressibility, the
tidal amplitude is within ~20% of the incompressible
case.

We have also evaluated the solution for the more complicated
case of multi-layered bodies. Such results are more relevant
to real layered planets, and address the question of whether
the instabilities that resulted in infinite Love numbers
were due to the artificially low densities near the core. We
find that the singularities do persist even for
multi-layered models, and even for cases that are not
extremely different from parameters for real planets.

This work was supported by NASA's Planetary Geology and
Geophysics program. \\ \\ [1] Love, A.E.H., Some Problems
of Geodynamics, New York Dover Publications, 1967 (reprint
of work done in 1911)

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Bulletin of the American Astronomical Society, **37** #2

© 2005. The American Astronomical Soceity.