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 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; also defined as the vertical (radial) displacement or variation of the planet's elastic properties. In terms of the tide generating potential formula_1, the displacement is formula_2 where formula_3 is latitude, formula_4 is east longitude and formula_5 is acceleration due to gravity. For a hypothetical solid Earth formula_6. For a liquid Earth, one would expect formula_7. However, the deformation of the sphere causes the potential field to change, and thereby deform the sphere even more. The theoretical maximum is formula_8. For the real Earth, formula_9 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 formula_10, where formula_11 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 formula_12, where formula_13 is the horizontal gradient operator. As with "h" and "k", formula_14 for a rigid body.
Values.
According to Cartwright, "An elastic solid spheroid will yield to an external tide potential formula_15 of spherical harmonic degree 2 by a surface tide formula_16 and the self-attraction of this tide will increase the external potential by formula_17." The magnitudes of the Love numbers depend on the rigidity and mass distribution of the spheroid. Love numbers formula_18, formula_19, and formula_20 can also be calculated for higher orders of spherical harmonics.
For elastic Earth the Love numbers lie in the range: formula_21, formula_22 and formula_23.
For Earth's tides one can calculate the tilt factor as formula_24 and the gravimetric factor as formula_25, where suffix two is assumed.