Love number

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The Love numbers (h, k, and l) are dimensionless parameters that measure the rigidity of a planetary body and the susceptibility of its shape to change in response to a tidal potential.

In 1909, Augustus Edward Hough Love introduced the values h and k which characterize the overall elastic response of the Earth to the tides ― Earth tides or body tides.[1] Later, in 1912, Toshi Shida added a third Love number, l, which was needed to obtain a complete overall description of the solid Earth's response to the tides.[2]

Definitions[edit]

The Love number h is defined as the ratio of the body tide to the height of the static equilibrium tide;[3] also defined as the vertical (radial) displacement or variation of the planet's elastic properties. In terms of the tide generating potential , the displacement is where is latitude, is east longitude and is acceleration due to gravity.[4] For a hypothetical solid Earth . For a liquid Earth, one would expect . However, the deformation of the sphere causes the potential field to change, and thereby deform the sphere even more. The theoretical maximum is . For the real Earth, lies between 0 and 1.

The Love number k is defined as the cubical dilation or the ratio of the additional potential (self-reactive force) produced by the deformation of the deforming potential. It can be represented as , where for a rigid body.[4]

The Love number l represents the ratio of the horizontal (transverse) displacement of an element of mass of the planet's crust to that of the corresponding static ocean tide.[3] In potential notation the transverse displacement is , where is the horizontal gradient operator. As with h and k, for a rigid body.[4]

Values[edit]

According to Cartwright, "An elastic solid spheroid will yield to an external tide potential of spherical harmonic degree 2 by a surface tide and the self-attraction of this tide will increase the external potential by ."[5] The magnitudes of the Love numbers depend on the rigidity and mass distribution of the spheroid. Love numbers , , and can also be calculated for higher orders of spherical harmonics.

For elastic Earth the Love numbers lie in the range: , and .[3]

For Earth's tides one can calculate the tilt factor as and the gravimetric factor as , where subscript two is assumed.[5]

References[edit]

  1. ^ Love Augustus Edward Hough. The yielding of the earth to disturbing forces 82 Proc. R. Soc. Lond. A 1909 http://doi.org/10.1098/rspa.1909.0008
  2. ^ TOSHI SHIDA, On the Body Tides of the Earth, A Proposal for the International Geodetic Association, Proceedings of the Tokyo Mathematico-Physical Society. 2nd Series, 1911-1912, Volume 6, Issue 16, Pages 242-258, ISSN 2185-2693, doi:10.11429/ptmps1907.6.16_242.
  3. ^ a b c "Tidal Deformation of the Solid Earth: A Finite Difference Discretization", S.K.Poulsen; Niels Bohr Institute, University of Copenhagen; p 24; [1] Archived 2016-10-11 at the Wayback Machine
  4. ^ a b c Earth Tides; D.C.Agnew, University of California; 2007; 174
  5. ^ a b Tides: A Scientific History; David E. Cartwright; Cambridge University Press, 1999, ISBN 0-521-62145-3; pp 140–141,224