Love number

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. 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.

Definitions

The Love number h is defined as the ratio of the body tide to the height of the static equilibrium tide; also defined as the vertical (radial) displacement or variation of the planet's elastic properties. In terms of the tide generating potential ${\displaystyle V(\theta ,\varphi )/g}$, the displacement is ${\displaystyle hV(\theta ,\varphi )/g}$ where ${\displaystyle \theta }$ is latitude, ${\displaystyle \varphi }$ is east longitude and ${\displaystyle g}$ is acceleration due to gravity. 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 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 ${\displaystyle kV(\theta ,\varphi )/g}$, where ${\displaystyle k=0}$ for a rigid body.

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. In potential notation the transverse displacement is ${\displaystyle l\mathrm{\nabla }(V(\theta ,\varphi ))/g}$, where ${\displaystyle \mathrm{\nabla }}$ is the horizontal gradient operator. As with h and k, ${\displaystyle l=0}$ for a rigid body.