Astronomy and Astrophysics – Astronomy
Scientific paper
Nov 2001
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2001dps....33.3702h&link_type=abstract
American Astronomical Society, DPS Meeting #33, #37.02; Bulletin of the American Astronomical Society, Vol. 33, p.1106
Astronomy and Astrophysics
Astronomy
Scientific paper
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 ρ , by solving the deformation equation [4], μ ∇ 2 u = ρ ∇U - (λ + μ ) ∇ (∇ ṡ u) where μ is the rigidity of the body and λ 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 u. A self-consistent solution can be obtained with U = \sum_{q=0}^{\infty} b_{(2+2q)} r^{(2+2q)} ( {3}/{2} \cos2 \theta - {1}/{2} ). In [1] and [3] only the r2 term was considered, which was valid only if compressibility is small or elasticity governs deformation (i.e. ρ g R << (λ + 2 μ )). The solution with only the r2 term reduces to Love's [1] solution in the limit of zero compressibility (λ = ∞ ). However, for rock μ ~ λ [4], in which case h2 is enhanced by ~ 3 %, and solutions for greater compressibility give up to 8 % enhancement of tidal amplitude. If ρ 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, Lunar Plan. Sci. XXXII 1741, 2001 [3] Hurford, T.A. and R. Greenberg, 2001 DDA meeting, Bull. Amer. Astron. Soc. in press [4] Kaula, W.M., John Wiley & Sons, Inc., 1968
Greenberg Richard
Hurford Terry A.
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