Love number

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 $V(\theta ,\phi )/g$ , the displacement is $hV(\theta ,\phi )/g$ where $\theta$ is latitude, $\phi$ is east longitude and $g$ is acceleration due to gravity.^[3] For a hypothetical solid Earth $h=0$ . For a liquid Earth, one would expect $h=1$ . However, the deformation of the sphere causes the potential field to change, and thereby deform the sphere even more. The theoretical maximum is $h=2.5$ . For the real Earth, $h$ 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 $kV(\theta ,\phi )/g$ , where $k=0$ for a rigid body.^[3]

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.^[2] In potential notation the transverse displacement is $l\nabla (V(\theta ,\phi ))/g$ , where $\nabla$ is the horizontal gradient operator. As with h and k, $l = 0$ for a rigid body.^[3]

According to Cartwright, "An elastic solid spheroid will yield to an external tide potential $U_{2}$ of spherical harmonic degree 2 by a surface tide $h_{2}U_{2}/g$ and the self-attraction of this tide will increase the external potential by $k_{2}U_{2}$ ."^[4] The magnitudes of the Love numbers depend on the rigidity and mass distribution of the spheroid. Love numbers $h_{n}$ , $k_{n}$ , and $l_{n}$ can also be calculated for higher orders of spherical harmonics.

For elastic Earth the Love numbers lie in the range: $0.616\leq h_{2}\leq 0.624$ , $0.304\leq k_{2}\leq 0.312$ and $0.084\leq l_{2}\leq 0.088$ .^[2]

For Earth's tides one can calculate the tilt factor as $1+k-h$ and the gravimetric factor as $1+h-(3/2)k$ , where suffix two is assumed.^[4]

References

↑ Marine Gravity, P. Dehlinger; Elsevier Scientific, 178, p 12
1 2 3 "Tidal Deformation of the Solid Earth: A Finite Difference Discretization", S.K.Poulsen; Niels Bohr Institute, University of Copenhagen; p 24;
1 2 3 Earth Tides; D.C.Agnew, University of California; 2007; 174
1 2 Tides: A Scientific History; David E. Cartwright; Cambridge University Press, 1999, ISBN 0-521-62145-3; pp 140–141,224

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[1] Marine Gravity, P. Dehlinger; Elsevier Scientific, 178, p 12

[Poulsen-2] 1 2 3 "Tidal Deformation of the Solid Earth: A Finite Difference Discretization", S.K.Poulsen; Niels Bohr Institute, University of Copenhagen; p 24;

[EarthTides-3] 1 2 3 Earth Tides; D.C.Agnew, University of California; 2007; 174

[Cartwright-4] 1 2 Tides: A Scientific History; David E. Cartwright; Cambridge University Press, 1999, ISBN 0-521-62145-3; pp 140–141,224