**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
[Previous] |
[Session 37] |
[Next]

## [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 h_{2} love number of a body
of radius R, density \rho, by solving the deformation
equation [4], \mu \nabla^{2} \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 = \sum_{q=0}^{\infty} b_{(2+2q)}
r^{(2+2q)} ( 3/2 \cos^{2} \theta - 1/2
). \end{equation}

In [1] and [3] only the r^{2} 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 r^{2} term reduces to Love's
[1] solution in the limit of zero compressibility (\lambda
= \infty). However, for rock \mu ~\lambda [4], in
which case h_{2} 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

[Previous] |
[Session 37] |
[Next]