Ivars Peterson's MathTrek
January 19, 2004
Tidal effects are caused by the gravitational force exerted on a satellite by its host planet and on the planet by its satellite. As the two bodies orbit each other, their surfaces rise and fall.
In essence, tides arise because a planet occupies a certain amount of space. Parts of the planet located at slightly different distances from a perturbing satellite experience a slightly different gravitational force. But because the planet remains intact, these parts are not entirely free to move separately.
Instead, the planet is deformed, developing internal strains and stresses that supply the forces necessary to make all its parts accelerate together in response to gravity's varying effect.
Ideally, a planet is stretched to produce a bulge that reaches its maximum extent along a line connecting the planet's center to that of the satellite. This bulge rotates at the same rate as the satellite orbits the planet. Satellites undergo analogous distortions.
In 1911, mathematician Augustus Edward Hough Love (1863–1940) of Oxford University formulated partial differential equations for calculating how the gravity of one body affects the shape of a second. One set provides the amount of deformation (tidal amplitude) for a compressible, uniform sphere.
Love's solution to this equation depends on only two factors. One is the ratio of a body's density times its radius times its surface gravity to its rigidity. The other is the ratio of a body's rigidity to its compressibility (as expressed by the so-called Lamé constant).
Love looked at two cases, using values for the constants that would be reasonable for characterizing Earth. He got plausible results for tidal amplitudes.
Terry A. Hurford, Sarah Frey, and Richard Greenberg of the University of Arizona have now taken a fresh look at Love's model. They used computers to study what would happen to the solutions over a wide range of radial, density, gravity, rigidity, and compressibility values.
For rocky bodies up to Earth's size, the solutions are well behaved and tidal amplitudes are reasonable. In certain other cases, however, the new calculations suggest that a sphere's shape can change drastically.
Applied to a planet only 20 percent or so larger than Earth or Venus, such extreme tides could lead to instability and, ultimately, disruption. Even the gravity of a body as small as a passing asteroid could induce large tidal effects.
"The possibility of frequent and substantial tidal deformation could have important implications for the evolution of such a planet," the researchers report. "Extrapolating further, it is conceivable that extreme instabilities might disrupt or limit the growth of forming planets."
It's possible that a growing planet, in the early stages of formation, could hit disruptive conditions when its density or radius or some other property reaches a critical value. Indeed, planet characteristics that allow large tides could signal a possible upper limit on planet size.
Depending on their physical characteristics, planetary satellites may also experience more extreme tides than scientists normally expect. Such effects could have important consequences for the thermal, geophysical, and orbital evolution of these satellites.
Interestingly, when a body's density and composition are not uniform but depend on its radius, tidal instabilities generally persist. Tidal amplitudes, however, tend to be much less extreme.
Copyright © 2004 by Ivars Peterson
Frey, S.E. 2004. Characterization of instabilities in the tidal deformation of a planetary body. Joint Mathematics Meetings. Jan. 7. Phoenix, Ariz. Abstract available at http://www.ams.org/amsmtgs/2078_abstracts/993-74-1144.pdf.
Hurford, T.A., S. Frey, and R. Greenberg. Preprint. Numerical evaluation of Love's solution for tidal amplitude: Extreme tides possible. Available at http://math.arizona.edu/~sfrey/Papers/tides02.pdf. Jensen, M.N. 2004. Old equation may shed new light on planet formation. University of Arizona news release. Jan. 5. Available at http://uanews.org/cgi-bin/WebObjects/UANews.woa/3/wa/SRStoryDetails?ArticleID=8407. Love, A.E.H. 1967. Some Problems of Geodynamics. Mineola, N.Y.: Dover. A biography of A.E.H. Love can be found at http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Love.html.
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A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.