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T.A. Hurford, R. Greenberg (Univ. of Arizona, LPL)
Most modern evaluations of tidal amplitude derive from 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 solve the more general case of arbitrary compressibility, which allows for a more straightforward derivation. We find the h2 love number of a body of radius R, density \rho, and surface gravity g to be \begin{equation}\label{eq:hcomp} h2 = \Bigg[\frac{5}{2}/1+\frac{19 \mu}{2 \rho g R}\Bigg] \Bigg{ 2 \rho g R (35+28\frac{\mu}{\lambda}) + 19 \mu (35+28\frac{\mu}{\lambda})/2 \rho g R (35+31\frac{\mu}{\lambda}) + 19 \mu (35+\frac{490}{19}\frac{\mu}{\lambda})\Bigg} \nonumber \end{equation} \lambda the Lamé constant. This h2 is the product of Love's expression for h2 (in square brackets) and a ``compressibility-correction'' factor (in {} brackets). Unlike Love's expression, this result is valid for any degree of compressibility (i.e. any \lambda). For the incompressible case (\lambda arrow \infty) the correction factor approaches 1, so that h2 matches the classical form given by Love.
In reality, of course, materials are not incompressible and the difference between our solution and Love's is significant. Assuming that the elastic terms dominate over the gravitational contribution (i.e. 19 \mu/(2 \rho g R) \gg 1), our solution can be ~7% percent larger than Love's solution for large \mu/\lambda. If the gravity dominates (i.e. 19 \mu/(2 \rho g R) \ll 1), our solution is ~10 % smaller than Love's solution for large \mu/\lambda. For example, a rocky body (\mu/\lambda ~ 1 [2]), Earth-sized (19\mu/(2 \rho g R) ~1) body, h2 would be reduced by about 1% from the classical formula. Similarly, under some circumstances the l2 Love number for a uniform sphere could be 22% smaller than Love's evaluation. \\ [1] Love, A.E.H., \emph{A Treatise on the Mathematical Theory of Elasticity}, New York Dover Publications, 1944 \\ [2] Kaula, W.M., \emph{An Introduction to Planetary Physics: The Terrestrial Planets}, John Wiley & Sons, Inc., 1968