DPS 2001 meeting, November 2001
Session 37. Galilean and Other Outer Planet Satellite Posters
Displayed, 9:00am Tuesday - 3:00pm Saturday, Highlighted, Friday, November 30, 2001, 9:00-10:30am, French Market Exhibit Hall

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[37.02] Tides on Self-gravitating, Compressible Bodies

T.A. Hurford, R. Greenberg (Univ. of Arizona, LPL)

Most modern derivations of tidal amplitude follow the approach presented by Love [1]. Love's analysis for a homogeneous sphere assumed an incompressible material, which required introduction of a non-rigorously justified pressure term. We have solved the more general case of arbitrary compressibility, which allows for a more straightforward derivation [2,3]. We find the h2 love number of a body of radius R, density \rho, by solving the deformation equation [4], \mu \nabla2 \vec{u} = \rho \vec{\nabla}U - (\lambda + \mu) \vec{\nabla}(\vec{\nabla} \cdot \vec{u}) where \mu is the rigidity of the body and \lambda the Lamé constant. The potential U is the sum of (a) the tide raising potential, (b) the potential of surface mass shifted above or below the spherical surface, (c) potential due to the internal density changes and (d) the change in potential of each bit of volume due to its displacement \vec{u}. A self-consistent solution can be obtained with

\begin{equation} U = \sumq=0\infty b(2+2q) r(2+2q) ( 3/2 \cos2 \theta - 1/2 ). \end{equation}

In [1] and [3] only the r2 term was considered, which was valid only if compressibility is small or elasticity governs deformation (i.e. \rho g R \ll (\lambda + 2 \mu)). The solution with only the r2 term reduces to Love's [1] solution in the limit of zero compressibility (\lambda = \infty). However, for rock \mu ~\lambda [4], in which case h2 is enhanced by ~3 %, and solutions for greater compressibility give up to 8 % enhancement of tidal amplitude.

If \rho g R is significant, higher order r(2q+2) terms are important and even greater corrections are required to the classical tidal amplitude. \\ \\ [1] Love, A.E.H., New York Dover Publications, 1944 [2] Hurford, T.A. and R. Greenberg, \emph{Lunar Plan. Sci. XXXII} 1741, 2001 [3] Hurford, T.A. and R. Greenberg, 2001 DDA meeting, \emph{Bull. Amer. Astron. Soc.} in press [4] Kaula, W.M., John Wiley & Sons, Inc., 1968

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