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 1911 (some authors have 1906)[1] Augustus Edward Hough Love introduced the values h and k which characterize the overall elastic response of the Earth to the tides. Later, in 1912, T. Shida of Japan added a third Love number, l, which was needed to obtain a complete overall description of the solid Earth's response to the tides.

The Love number h is defined as the ratio of the body tide to the height of the static equilibrium tide;[2] 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.[3] 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 these values.

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.[3]

The Love number l represents the ratio of the horizontal (transverse) displacement of as element of mass of the planet's crust to that of the corresponding static ocean tide.[2] In potential notation the transverse displacement is , where is the horizontal gradient operator. As with h and k, for a rigid body.[3]

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 ."[4] 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 .[2]

For Earth's tides one can calculate the tilt factor as and the gravimetric factor as , where suffix two is assumed.[4]


  1. ^ Marine Gravity, P. Dehlinger; Elsevier Scientific, 178, p 12
  2. ^ 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]
  3. ^ a b c Earth Tides; D.C.Agnew, University of California; 2007; 174
  4. ^ a b Tides: A Scientific History; David E. Cartwright; Cambridge University Press, 1999, ISBN 0-521-62145-3; pp 140–141,224