Secular tidal changes in lunar orbit and Earth rotation


Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = −0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.

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We thank the lunar laser ranging stations at McDonald Observatory, Texas, Observatoire de la Côte d’Azur, France, Haleakala Observatory, Hawaii, Apache Point Observatory, New Mexico, and Matera, Italy that provided the data sets that make LLR analyses possible. LLR data are available from the International Laser Ranging Service archive at We acknowledge extensive conversations with D. Pavlov about tidal modeling that benefited this paper. C. F. Yoder contributed to the early development of the solar perturbation scaling factors for LLR results. M. Efroimsky provided a valuable review. The research described in this paper was carried out at the Jet Propulsion Laboratory of the California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Government sponsorship acknowledged.

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List of symbols

a :

Semimajor axis of lunar orbit

\(a'\) :

Semimajor axis of solar orbit

C :

Moment of inertia

\(C_{2m}\) :

Gravity field coefficients

D :

Mean elongation of Moon from Sun

e :

Eccentricity of lunar orbit

F :

Lunar mean argument of latitude

G :

Gravitational constant

\(h_{2}\) :

Vertical Love number

i :

Lunar inclination to ecliptic plane

\(k_{2}\) :

Potential Love number

\(K_{V}\) :

Parameter for dissipation at lunar CMB

l :

Lunar mean anomaly

\(l'\) :

Solar mean anomaly

L :

Lunar mean longitude

\(L'\) :

Solar mean longitude

m :


\(M_{E}\) :

Mass of Earth

\(M_{M}\) :

Mass of Moon

\(M_{S}\) :

Mass of Sun

\(M'\) :

Mass of external body

n :

Sidereal mean motion of Moon

\(n'\) :

Sidereal mean motion of Sun

P :


\(P_{2m}\) :

Associated Legendre polynomial

q :

Index for different periods in Fourier series

r :

Distance from center of Earth to body

R :

Radius of Earth

S :

Scaling parameters near unity

\(S_{2m}\) :

Gravity field coefficient

t :


\(u_{i}\) :

Unit vector from Earth to external body

\(U_{ij }\) :

Functions (a/\(r)^{3} u_{i} u_{j}\)

\(U_{ijq }\) :

Periodic term of \(U_{ij}\)

\(V_{2}\) :

Potential from tidal distortion

\(W_{2}\) :

Tide raising potential

\(\alpha \) :

Right ascension

\(\varepsilon \) :


\(\Delta \) :

Small difference

\(\Delta C_{2m}\) :

Degree-2 tidal gravity field coefficients

\(\Delta \) S \(_{2m}\) :

Degree-2 tidal gravity field coefficients

\(\varTheta \) :

Earth-centered angle between an external body and a selected point

\(\vartheta \) :

Rotation angle between precessing equinox and zero longitude

\(\lambda \) :

Terrestrial longitude

\(\tau _{0}\) :

Zonal time delay

\(\tau _{1}\) :

Diurnal time delay for orbit

\(\tau _{2}\) :

Semidiurnal time delay for orbit

\(\tau _{R1}\) :

Diurnal time delay for rotation

\(\tau _{R2}\) :

Semidiurnal time delay for rotation

\(\tau _{M}\) :

Time delay for tides on Moon

\(\chi \) :

Phase lag

\(\omega \) :

Lunar mean argument of perigee

\(\omega _{E}\) :

Spin rate of Earth

\(\varOmega \) :

Lunar mean node

\(\varpi \) :

Lunar longitude of perigee \(\varOmega \)+\(\omega \)

\(\zeta \) :

Angular argument

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Williams, J.G., Boggs, D.H. Secular tidal changes in lunar orbit and Earth rotation. Celest Mech Dyn Astr 126, 89–129 (2016).

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  • Tides
  • Lunar orbit
  • Earth rotation
  • Tidal acceleration
  • Tidal dissipation
  • Moon
  • Lunar laser ranging (LLR)