Love number
Encyclopedia
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) 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 tide
s.
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 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. For a hypothetical solid Earth h = 0 and h = 1 for a liquid Earth. For the real Earth, 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 k V(θ, φ)/g . Where k = 0 for a rigid body.
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. 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.
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." The magnitudes of the Love numbers depend on the rigidity and mass distribution of the spheroid. Love numbers hn, kn, and ln 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
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.
Dimensionless quantity
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...
parameters that measure the rigidity
Rigidity
Rigid or rigidity may refer to:*Stiffness, the property of a solid body to resist deformation, which is sometimes referred to as rigidity*Structural rigidity, a mathematical theory of the stiffness of ensembles of rigid objects connected by hinges...
of a planetary body and the susceptibility of its shape to change in response to a tidal potential
Tidal force
The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational force per unit mass exerted on one body by a second body is not constant across its diameter, the side nearest to the second being more attracted by it than the side...
.
In 1911 (some authors have 1906) Augustus Edward Hough Love
Augustus Edward Hough Love
Augustus Edward Hough Love FRS , often known as A. E. H. Love, was a mathematician famous for his work on the mathematical theory of elasticity...
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 tide
Tide
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the moon and the sun and the rotation of the Earth....
s.
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 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. For a hypothetical solid Earth h = 0 and h = 1 for a liquid Earth. For the real Earth, 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 k V(θ, φ)/g . Where k = 0 for a rigid body.
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. In potential notation the transverse displacement is l del(V(θ, φ))/g, where del is the horizontal gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
operator. As does h and k, l = 0 for a rigid body.
According to Cartwright, "An elastic solid spheroid will yield to an external tide potential U2 of spherical harmonic
Spherical Harmonic
Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
degree 2 by a surface tide h2U2/g and the self-attraction of this tide will increase the external potential by k2U2." The magnitudes of the Love numbers depend on the rigidity and mass distribution of the spheroid. Love numbers hn, kn, and ln 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
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.