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 1909 [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.

Definitions

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

Values

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

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

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

References

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. "Tidal Deformation of the Solid Earth: A Finite Difference Discretization", S.K.Poulsen; Niels Bohr Institute, University of Copenhagen; p 24; Archived 2016-10-11 at the Wayback Machine
3. Earth Tides; D.C.Agnew, University of California; 2007; 174
4. Tides: A Scientific History; David E. Cartwright; Cambridge University Press, 1999, ISBN 0-521-62145-3; pp 140–141,224