Love number

The Love numbers (h, k, and l) are dimensionless parameters that measure the rigidity of a planetary body or other gravitating object, and the susceptibility of its shape to change in response to an external 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 ― Earth tides or body tides. ^[1] 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. ^[2]

Definitions

The Love number h is defined as the ratio of the body tide to the height of the static equilibrium tide; ^[3] also defined as the vertical (radial) displacement or variation of the planet's elastic properties. In terms of the tide generating potential ${\displaystyle V(\theta ,\phi )/g}$, the displacement is ${\displaystyle hV(\theta ,\phi )/g}$ where ${\displaystyle \theta }$ is latitude, ${\displaystyle \phi }$ is east longitude and ${\displaystyle g}$ is acceleration due to gravity. ^[4] 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 ${\displaystyle kV(\theta ,\phi )/g}$, where ${\displaystyle k=0}$ for a rigid body. ^[4]

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. ^[3] In potential notation the transverse displacement is ${\displaystyle l\nabla (V(\theta ,\phi ))/g}$, where ${\displaystyle \nabla }$ is the horizontal gradient operator. As with h and k, ${\displaystyle l=0}$ for a rigid body. ^[4]

Values

According to Cartwright, "An elastic solid spheroid will yield to an external tide potential ${\displaystyle U_{2}}$ of spherical harmonic degree 2 by a surface tide ${\displaystyle h_{2}U_{2}/g}$ and the self-attraction of this tide will increase the external potential by ${\displaystyle k_{2}U_{2}}$." ^[5] The magnitudes of the Love numbers depend on the rigidity and mass distribution of the spheroid. Love numbers ${\displaystyle h_{n}}$, ${\displaystyle k_{n}}$, and ${\displaystyle l_{n}}$ can also be calculated for higher orders of spherical harmonics.

For elastic Earth the Love numbers lie in the range: ${\displaystyle 0.616\leq h_{2}\leq 0.624}$, ${\displaystyle 0.304\leq k_{2}\leq 0.312}$ and ${\displaystyle 0.084\leq l_{2}\leq 0.088}$. ^[3]

For Earth's tides one can calculate the tilt factor as ${\displaystyle 1+k-h}$ and the gravimetric factor as ${\displaystyle 1+h-(3/2)k}$, where subscript two is assumed. ^[5]

Neutron stars are thought to have high rigidity in the crust, and thus a low Love number; ${\displaystyle 0.05\leq k_{2}\leq 0.17}$, ^{[6] [7]} while black holes have vanishing Love numbers for all multipoles ${\displaystyle k_{\ell }=0}$. ^{[8] [9] [10]} Measuring the Love numbers of compact objects in binary mergers is a key goal of gravitational-wave astronomy.

References

1. Love Augustus Edward Hough. The yielding of the earth to disturbing forces 82 Proc. R. Soc. Lond. A 1909 http://doi.org/10.1098/rspa.1909.0008
2. 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, doi : 10.11429/ptmps1907.6.16_242.
3. "Tidal Deformation of the Solid Earth: A Finite Difference Discretization", S.K.Poulsen; Niels Bohr Institute, University of Copenhagen; p 24; Archived 2016-10-11 at the Wayback Machine
4. Earth Tides; D.C.Agnew, University of California; 2007; 174
5. Tides: A Scientific History; David E. Cartwright; Cambridge University Press, 1999, ISBN   0-521-62145-3; pp 140–141,224
6. Yazadjiev, Stoytcho S.; Doneva, Daniela D.; Kokkotas, Kostas D. (October 2018). "Tidal Love numbers of neutron stars in f(R) gravity". The European Physical Journal C. 78 (10): 818. arXiv: 1803.09534 . Bibcode:2018EPJC...78..818Y. doi:10.1140/epjc/s10052-018-6285-z. PMC   6244867 . PMID   30524193.
7. Hinderer, Tanja; Lackey, Benjamin D.; Lang, Ryan N.; Read, Jocelyn S. (23 June 2010). "Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral". Physical Review D. 81 (12): 123016. arXiv: 0911.3535 . Bibcode:2010PhRvD..81l3016H. doi:10.1103/PhysRevD.81.123016. hdl: 1721.1/64461 . S2CID   14819350.
8. Damour, Thibault; Nagar, Alessandro (2009-10-23). "Relativistic tidal properties of neutron stars". Physical Review D. 80 (8). arXiv: 0906.0096 . doi:10.1103/PhysRevD.80.084035. ISSN   1550-7998.
9. Binnington, Taylor; Poisson, Eric (2009-10-14). "Relativistic theory of tidal Love numbers". Physical Review D. 80 (8): 084018. arXiv: 0906.1366 . doi:10.1103/PhysRevD.80.084018.
10. Chia, Horng Sheng (2021-07-06). "Tidal deformation and dissipation of rotating black holes". Physical Review D. 104 (2). arXiv: 2010.07300 . doi:10.1103/PhysRevD.104.024013. ISSN   2470-0010.