The Love numbers (''h'', ''k'', and ''l'') are
dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
parameters that measure the
rigidity of a
planetary body
A planetary-mass object (PMO), planemo, or planetary body is by geophysical definition of celestial objects any celestial object massive enough to achieve hydrostatic equilibrium (to be rounded under its own gravity), but not enough to sustain ...
and the susceptibility of its shape to change in response to a
tidal potential.
In 1909,
Augustus Edward Hough Love
Augustus Edward Hough Love FRS (17 April 1863, Weston-super-Mare – 5 June 1940, Oxford), often known as A. E. H. Love, was a mathematician famous for his work on the mathematical theory of elasticity. He also worked on wave propagation and h ...
introduced the values ''h'' and ''k'' which characterize the overall
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
response of the Earth to the tides ― ''
Earth tide
Earth tide (also known as solid Earth tide, crustal tide, body tide, bodily tide or land tide) is the displacement of the solid earth's surface caused by the gravity of the Moon and Sun. Its main component has meter-level amplitude at periods of a ...
s'' 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
tide
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another.
Tide tables ...
s.
[TOSHI SHIDA, On the Body Tides of the Earth, A Proposal for the International Geodetic Association, Proceedings of the Tokyo Mathematico-Physical Society. 2nd Series, 1911-1912, Volume 6, Issue 16, Pages 242-258, ISSN 2185-2693, .]
Definitions
The Love number ''h'' is defined as the ratio of the body tide to the height of the static
equilibrium tide;
["Tidal Deformation of the Solid Earth: A Finite Difference Discretization", S.K.Poulsen; Niels Bohr Institute, University of Copenhagen; p 24]
also defined as the vertical (radial) displacement or variation of the planet's elastic properties. In terms of the tide generating potential
, the displacement is
where
is latitude,
is east longitude and
is acceleration due to gravity.
[Earth Tides; D.C.Agnew, University of California; 2007; 174] For a hypothetical solid Earth
. For a liquid Earth, one would expect
. However, the deformation of the sphere causes the potential field to change, and thereby deform the sphere even more. The theoretical maximum is
. For the real Earth,
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
, where
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 , where is the horizontal ]gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
operator. As with ''h'' and ''k'', for a rigid body.[
]
Values
According to Cartwright, "An elastic solid spheroid will yield to an external tide potential of spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form ...
degree 2 by a surface tide and the self-attraction of this tide will increase the external potential by ."[Tides: A Scientific History; David E. Cartwright; Cambridge University Press, 1999, ; pp 140–141,224] The magnitudes of the Love numbers depend on the rigidity and mass distribution of the spheroid. Love numbers , , and can also be calculated for higher orders of spherical harmonics.
For elastic Earth the Love numbers lie in the range: , and .[
For Earth's tides one can calculate the tilt factor as and the gravimetric factor as , where subscript two is assumed.][
]
References
{{reflist
Tides
Elasticity (physics)
Dimensionless numbers of mechanics
Geodynamics