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)
Bulletin of the American Astronomical Society, 37 #2
© 2005. The American Astronomical Soceity.