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(θ, φ)/g, the displacement is h V(θ, φ)/g where θ is latitude, φ is east longitude and g is acceleration 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 k V(θ, φ)/g . Where k = 0 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 l del(V(θ, φ))/g, where del is the horizontal gradient operator. As does h and k, l = 0 for a rigid body.^[3]

According to Cartwright, "An elastic solid spheroid will yield to an external tide potential U₂ of spherical harmonic degree 2 by a surface tide h₂U₂/g and the self-attraction of this tide will increase the external potential by k₂U₂."^[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 ≤ h₂ ≤ 0.624, 0.304 ≤ k₂ ≤ 0.312 and 0.084 ≤ l₂ ≤ 0.088 ^[2]

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

References