Williams et al. Planetary Science 2014, 3:2
http://www.planetary-science.com/content/3/1/2
ORIGINAL RESEARCH
Open Access
The past and present Earth-Moon system: the
speed of light stays steady as tides evolve
James G Williams*, Slava G Turyshev and Dale H Boggs
* Correspondence:
james.g.williams@jpl.nasa.gov
Jet Propulsion Laboratory, California
Institute of Technology, Pasadena,
CA 91109, USA
Abstract
Tides induce a semimajor axis rate of +38.08 ± 0.19 mm/yr, corresponding to an
acceleration of the Moon’s orbital mean longitude of −25.82 ± 0.13 "/cent2, as
determined by the analysis of 43 yr of Lunar Laser Ranging (LLR) data. The LLR
result is consistent with analyses made with different data spans, different analysis
techniques, analysis of optical observations, and independent knowledge of tides.
Plate motions change ocean shapes, and geological evidence and model calculations
indicate lower rates of tidal evolution for extended past intervals. Earth rotation has
long-term slowing due to tidal dissipation, but it also experiences variations for times
up to about 105 yr due to changes in the moment of inertia. An analysis of LLR data
also tests for any rate of change in either the speed of light c or apparent mean
distance. The result is (−2.8 ± 3.4)×10–12 /yr for either scale rate or –(dc/dt)/c, or
equivalently −1.0 ± 1.3 mm/yr for apparent distance rate. The lunar range does not
reveal any change in the speed of light.
Keywords: Moon; Tides; Lunar tidal acceleration; Lunar orbit evolution; Speed of
light; Lunar laser ranging; Earth rotation
PACS numbers: 06.30.Gv Measurements common to several branches of physics
and astronomy; Velocity; 91.10.Tq Solid earth physics; Geodesy and gravity; Earth
tides; 96.12.De Solar system; Solid surface planets; Orbital and rotational dynamics
Background
The gravitational attraction of the Moon at the Earth causes a tidal distortion of the
oceans and solid Earth. The tidal bulges are not quite aligned with the direction to the
Moon. The orientation of the bulges leads the direction to the Moon, because a
delayed response is carried forward by Earth rotation. There results a forward acceleration on the Moon and a deceleration of the Earth’s spin; energy and angular momentum are transferred from the Earth to the lunar orbit. The Moon’s mean distance
increases and its orbit period increases. Analysis of accurate laser measurements of the
range between observatories on the Earth and retroreflectors on the Moon determines
these orbit changes with 1/2% accuracy. The “Calculation of lunar orbit anomaly”
paper by Riofrio [1] claims that the present 38.1 mm/yr Lunar Laser Ranging (LLR)
value for the tidal recession of the Moon is 9–10 mm/yr too large because a decreasing
speed of light causes an apparent increase in distance. To support this idea, evidence
from geology, Earth rotation, and ocean model calculations are cited. We discuss the
tide-related geological and geophysical evidence. We outline the LLR sensitivity to
© 2014 Williams et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Williams et al. Planetary Science 2014, 3:2
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both lunar tidal recession rate and increasing orbit period (0.352 ms/yr), and compare
the recent values with older determinations. Analysis of LLR data also tests whether
the speed of light is constant.
Results and discussion
Lunar tidal acceleration and recession rate
The transfer of energy and angular momentum from the rotation of the Earth to the
orbit of the Moon causes the length of day, the lunar distance, and the lunar orbit
period to increase. By deducing this mechanism, tidal recession was predicted theoretically by George Darwin in the late 19th century prior to its detection [2,3]. By convention, the tidal increase in the lunar orbit period is presented as a tidal decrease in mean
motion that is equal to a negative tidal acceleration in orbital longitude. Tidal acceleration was detected and presented by Spencer Jones [4] and Clemence [5] in the last
century from the analysis of optical observations of the Moon, Sun, and planets. According to Kepler’s third law, a negative acceleration in orbital mean longitude (mean
longitude = mean anomaly + argument of perigee + node, and its derivative is mean
motion) corresponds to a linear increase in semimajor axis. Tidal acceleration is frequently denoted by dn/dt and semimajor axis rate by da/dt. The third-law connection
2a dn/dt + 3n da/dt = 0 gives a first approximation. Some selected values follow:
1) Morrison and Ward [6] found a lunar tidal acceleration in longitude of −26 ± 2
seconds of arc/century2 (“/cent2) from the analysis of optical observations. Analysis
of timings of transits of Mercury across the Sun from 1677–1973 allowed the
changing angular rotation of the Earth to be determined separately from the lunar
orbital tidal acceleration. Before accurate clocks became available in the middle of
the last century, Earth rotation was a “clock” for celestial observations. The
deceleration and other variations in Earth rotation needed to be determined with
respect to a uniform physical time scale in order to determine the lunar tidal
acceleration with respect to that time scale. The orbit of Mercury provided the
uniform time scale.
2) Dickey et al. [7] determined a lunar tidal acceleration in longitude of −25.88 ±
0.5 “/cent2 and a semimajor axis rate of +38.2 ± 0.7 mm/yr from analysis of
24 yr of Lunar Laser Ranging (LLR) data. The lunar orbit and orientation were
generated by numerical integration.
3) Chapront et al. [8] found a tidal acceleration of −25.858 “/cent2 from analysis of
LLR data. They used series representations for lunar orbit and orientation.
4) Williams et al. [9,10] obtained a tidal acceleration of −25.85 “/cent2 and a
semimajor axis rate of +38.14 mm/yr for the DE421 lunar ephemeris, which was
derived from analysis of 38 yr of LLR data. DE421 was integrated numerically.
5) Williams et al. [11] determine a tidal acceleration of −25.82 ± 0.13 “/cent2 and a
semimajor axis rate of +38.08 ± 0.19 mm/yr for the recent DE430 lunar ephemeris,
which is derived from analysis of 43 yr of LLR data (18,548 ranges from March
1970 to December 2012).
The agreement between the Morrison and Ward [6] result and the LLR analyses
[7-11] demonstrates that optical observations, mainly occultation timings, and laser
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ranges detect the same tidal acceleration. The tidal acceleration value derived by Chapront
et al. [8] shows that an independent approach and software yields compatible LLR results.
The JPL results [7,9-11] demonstrate that with increasing data span and improving uncertainty the tidal acceleration determination is stable.
Ordered by decreasing contribution, the M2, O1, and N2 tides determine most of the
tidal acceleration. But for eccentricity rate the order is N2, Moon tides, Q1, and M2,
with the second and last being negative. The tide model for DE421 adjusted one timedelay dissipation parameter for semidiurnal tides, one for diurnal tides, and one for
Moon tides. These parameters fit the tidal acceleration well, but were less successful
with the eccentricity rate [10]. The model for tidal perturbations from tides on the
Earth was improved prior to DE430 [11] to allow for time-delay shifts across the
diurnal and semidiurnal bands. The DE430 tidal eccentricity rate is 1.4×10–11/yr, with
1.8×10–11/yr coming from tides on the Earth and −0.4×10–11/yr from solid-body tides
on the Moon. This is an improvement over DE421 (0.9×10–11/yr), but when an analytical eccentricity rate solution parameter is added to post-DE430 LLR analyses, then an
additional rate is found and the total eccentricity rate becomes (1.9 ± 0.2)×10–11/yr.
The extra eccentricity rate, in addition to the rate from our tidal model, implies that
further dissipation-related modeling improvements are possible.
There is further evidence supporting the modern determinations of lunar tidal acceleration and recession. Tides on the Earth have been studied with satellites. Altimetry
measures ocean tide heights, and tidal attraction is determined from perturbations on
satellite orbits. These studies separate the tides into different periodic components. For
example, the largest semidiurnal tide is the M2 tide with a period of 12.42 hr, and the
largest diurnal tide is the O1 tide with a 25.82-hr period. There are also slow zonal
tides with periods of one month and one-half month. Tides raised by the Sun are about
half the size of tides raised by the Moon. Lunar tidal acceleration is mainly caused by
the gravity from Moon-raised tides on the Earth acting back on the Moon. Lunar tidal
acceleration, computed from the satellite-determined tidal components presented by
Lyard et al. [12] and Ray [13], compared favorably with the tidal acceleration of
the LLR-derived DE421 lunar orbit [9,10]. Williams et al. [10] found a difference
of less than 1% between the two values. The DE430 tidal acceleration [11] also
agrees within 1%.
The lunar tidal anomaly paper [1] focuses its discussion on the semimajor axis rate
da/dt, rather than the acceleration in orbital mean longitude. The LLR data analyses
are more sensitive to the acceleration in mean anomaly, a near proxy for the acceleration in mean longitudea, than to the recession rate because the lunar orbit is eccentric
[14]. Because there is a monthly variation of the radius of the lunar orbit, any perturbation of mean anomaly will cause a perturbation in that radius. For example, the tidal
acceleration causes a −2.4 m/yr2 t2 perturbation in the product of semimajor axis a and
mean longitude perturbation, and an approximate +0.038 m/yr t – 0.13 m/yr2 t2 sin
(mean anomaly) perturbation in radius, where the time t in years is zero at the epoch
when the perturbation starts to accumulate. The oscillating t2 term in radius is much
stronger than the linear t term from the semimajor axis for times of years to decades. By only considering the increasing semimajor axis, [1] incorrectly assumed
that LLR data analysis would mistake any constant rate in radius for a tidal perturbation. The perturbation in orbital radius from the DE430 tidal eccentricity rate
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is approximately −0.005 m/yr t cos(mean anomaly), distinct from either a linear increase
or a t2 sin(mean anomaly). The foregoing expressions are for illustration; the LLR
programs use an integrated orbit.
The moon’s evolving orbit and the earth’s decelerating spin
The lunar anomaly paper [1] offered three types of evidence in support of a slower
semimajor axis rate: geological evidence from a rhythmite, ocean model calculations of
tides, and nontidal acceleration of the Earth’s rotation.
Rhythmites
Tidal rhythmites preserve geological layering from ancient tides, and they may allow
the past evolution of the Moon to be unraveled. Modulation of the layers may permit
the number of days per month, days per year, or months per year to be determined
after identifying the cause of each periodicity. The lunar anomaly paper [1] selected the
310 million year old Mansfield sediment to derive a 2.9 ± 0.6 cm/yr average lunar recession rate over the 310 million year interval. A review of tidal rhythmites and related
structures is presented by Coughenur et al. [15]. This review gives a 2.17 ± 0.31 cm/yr
average recession rate for the 620 million year old Reynella Siltstone, a member of the
Elatina Formation. We accept the idea that the past rate of lunar recession was lower
than the present value for extended spans of time, but we do not accept the practice of
using a past rate to replace or assess the accuracy of the current rate.
Most of the tidal dissipation that causes the recession of the Moon occurs in the
oceans. The pattern of each tidal component is complicated; see Coughenur et al. [15]
and Poliakow [16] for examples. Each tidal component has an individual period, and
for each the pattern of local tide heights and phases depends on location. The combination of tidal components also depends on location, a complication for the analysis of
rhythmites. The pattern for each tidal component can be combined into a global series
of spherical harmonic functions. In a global sense, energy dissipation causes each tidal
component to shift orientation with respect to the Moon’s attraction. This phaseshifted part of each tidal component causes lunar tidal acceleration, semimajor axis
rate, and eccentricity rate. Plate motion changes the shapes and locations of the oceans,
and this causes substantial variations in the tidal acceleration over ~108 yr time scales
[16]. Although the tidal acceleration varies over these long time scales, the Moon
continues to move outward.
Tide models
Poliakow [16] computed the past evolution of one tidal component, M2, the largest
semidiurnal component. The M2 tidal component contributes ~80% of the current
total tidal acceleration. Although the M2 tide does not cause the total recession rate,
the lunar anomaly paper [1] cites the M2 calculation of a current +29 mm/yr contribution [16] as though the M2 rate was the total recession rate. For comparison, the
DE430 lunar ephemeris has a 31 mm/yr recession rate caused by the M2 component.
Other DE430 rates are +33.5 mm/yr from all semidiurnal tides, +5.1 mm/yr from all diurnal tides, and −0.5 mm/yr from zonal tides on the Earth and tides on the Moon [11].
Although [1] cites Poliakow [16] for support for a 29 mm/yr recession rate, it contradicts his calculations for large variation in the past M2-caused rate by saying “For the
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Moon’s recession to vary so greatly, tidal heights would have to increase enormously
over time.” In addition to contradicting the cited paper, this statement seems to confuse
the roles of local tide heights that influence rhythmites and phase-shifted components
of global tides that cause the lunar recession rate. Any claim of steady tide heights
based on rhythmites must be viewed with skepticism.
Over 108 yr time scales, the oceans changed shape due to plate motion, which
affected tides. Over even longer time scales, the more rapidly spinning Earth of the past
shifted the tidal frequencies with respect to the resonant (normal mode) frequencies of
the oceans. Bills and Ray [17] considered several models of changing ancient tides
including models of Webb [18] and Hansen [19]. Those studies and [16] found that
past tidal dissipation varied by large amounts. Although Bills and Ray concluded that
present-day tidal dissipation was more effectiveb than in the past, they considered the
reason to be understood. Despite the understanding demonstrated by [16-19], [1] implies that the Bills and Ray work considered differences between past and present tidal
recession rates to be an anomaly. Tides evolve due to plate motion and slowing
spin rate.
Nontidal acceleration of Earth rotation
The increasing orbital angular momentum of the Moon’s evolving orbit is supplied by a
decreasing terrestrial spin angular momentum. Determinations of the deceleration of
Earth rotation by Stephenson and Morrison [20] yield about 3/4 of the deceleration expected from lunar tidal accelerationc. The difference is called nontidal acceleration. The
cause of the nontidal acceleration of Earth rotation was explained three decades ago by
Yoder et al. [21]. They found that the Earth’s oblate shape and moment of inertia C are
decreasing. Since spin angular momentum Cω is the product of the Earth’s moment
and spin rate ω, a negative dC/dt causes a positive contribution to dω/dt since dω/dt =
[T – ω dC/dt ]/C, where the tidal torque T is negative. From the analysis of satellite
tracking data, the Earth’s dJ2/dt was determined to be negative by [21], where J2 is the
degree-2 coefficient of the gravitational potential that describes the oblateness of the
Earth’s gravity field. Since dC/dt = (2/3) MR2 dJ2/dt, where M is the mass and R is the
equatorial radius, the change in moment C is established. The nontidal acceleration of
rotation from (dω/dt)NT = −ω (dC/dt)/C agrees with the historical nontidal acceleration
from [20].
The Earth’s moment of inertia and rotation are affected by short- and long-term effects. Short-term effects can include ocean and atmospheric mass redistribution, elastic
response to changing loads, and a tide that changes the moment of inertia with an
18.6-yr period. A recent Cheng et al. [22] analysis finds a negative J2 rate from 1976
to ~1995 that agrees with Yoder et al. [21], but the rate subsequently decreases,
possibly due to (short-term) modern global warming and the resulting redistribution of
water mass. Stephenson and Morrison [20] analyzed Earth rotation over 2700 yr, so
long-term changes apply. The initial determination of dJ2/dt led [21] to an explanation
for the nontidal acceleration of the Earth’s rotation. The negative dC/dt was interpreted
to be from viscous rebound of the Earth following deglaciation near the end of the last
ice age ~104 yr ago. The weight of glaciers depressed the surface of the Earth under the
polar ice during the ice ages, and after melting removed that weight the Earth’s surface
started rebounding upward. Viscous rebound is slow lasting thousands of years and
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continuing today. The slow shape change from viscous rebound causes a nontidal
acceleration of Earth rotation with a sign opposite to that of tidal deceleration. Over
the longer ~105 yr time scale of ice age cycles, the nontidal acceleration changes sign
and is not truly secular. The past decrease of J2 has been detected, and the directly
linked nontidal acceleration of rotation applies to historical data.
Under Possible explanations, the lunar anomaly paper [1] attempts to mention and
discount the viscous rebound explanation, but the statements there are confusing. The
viscous rebound of the Earth has an exponential relaxation time of several thousand
years; large-scale deglaciation is only required near the end of the last (quaternary) ice
age ~10–15 thousand years ago, so extensive deglaciation is not required for historical
times. Nontidal acceleration of Earth rotation does not change tidal friction, but it complicates any use of the deceleration of Earth rotation to infer lunar tidal recession,
which [1] attempts.
Testing whether the speed of light decreases
The lunar anomaly paper [1] proposes that the speed of light c is slowing with time. Although a slowing speed of light would cause an increase in the apparent lunar distance,
it would not change the tidal acceleration in orbital longitude, already conflicting with
the observational results given earlier. Still, an apparent nontidal increase in distance or
scale is a testable prediction. LLR data were analyzed to seek any rate of change of the
round-trip time of the laser pulse, the “range,” that was distinct from lunar tidal
acceleration and recession [23]. Apart from tidal recession, [23] found a limit for the
absolute value of any anomalous distance rate of <3.5 mm/yr, a limit that converts
to |scale rate| = |(dc/dt)/c| <0.9×10–11 /yr. This limit is smaller than the prediction in
[1] of −2.4×10–11/yr for (dc/dt)/c, or +9 mm/yr in apparent distance. Although [1] cited
the LLR paper on relativity [23], it did not mention this result that contradicts the
dc/dt prediction. That earlier solution [23] is updated here: we fit 18,696 laser ranges
between March 1970 and April 2013; in addition to scale rate, LLR solution parameters
include diurnal and semidiurnal tidal acceleration parameters, tidal dissipation in the
Moon, an eccentricity rate in addition to that caused by our tidal model, parameters
for lunar orbit (including mean distance) and orientation, locations of ranging
stations and retroreflector arrays, and other standard LLR solution parameters [11].
Folkner et al. [24] detail the formulation for lunar orbit and orientation. For scale
rate, or –(dc/dt)/c, we obtain (−2.8 ± 3.4)×10–12 /yr, or −1.0 ± 1.3 mm/yr in apparent distance. This test result is much smaller than the dc/dt prediction of [1]. The correlations
between the diurnal and semidiurnal tidal acceleration parameters and the scale rate parameter are small, –0.03 and +0.03, respectively, supporting the earlier assertion that a t2
sin(mean anomaly) perturbation of orbital radius is distinct from a linear increase. The
correlation between eccentricity rate and scale rate is −0.14, and its correlations with the
two tidal acceleration parameters are −0.05 and +0.19, respectively. There is a good separation of parameters.
The age of the expanding universe is 1.38×1010 yr. The scale rate computed from the
inverse age is 0.72×10–10 /yr; this is the Hubble constant expressed with different units
than the usual km/s/Mpc. The lunar semimajor axis rate gives (da/dt)/a = 0.99×10–10 /yr,
where a = 384,399 km. This similarity of numbers led Van Flandern [25,26], before Riofrio
[1], to attempt to link the lunar recession rate to cosmology. He proposed that tidal
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acceleration would be different for atomic and dynamical time scales, and the time scale
difference would be caused by a decreasing gravitational constant G that was linked to the
Hubble constant. Modern results do not support either a difference in time scales, e.g.,
the agreement of [6] with [7-11], or a changing G [23,27,28]. The similar values of
(da/dt)/a and the Hubble constant are due to multi-billion year ages for the EarthMoon system and the universe. The Earth, Moon, and solar system are ~4.55×109 yr old.
The lunar (da/dt)/a must be smaller than the inverse age of the Earth-Moon system since
the tidal recession rate was faster when the Moon was close to the Earth. The solar system
age is about 1/3 of the age of the universe, so the similarity of the two rates does not
require an unusual explanation. The lunar range provides no observational evidence for a
slowing of the speed of light, or any other connection between the apparent lunar recession rate and the age of the universe.
Conclusions
Present day lunar tidal acceleration values are consistent between optical occultation
timings and laser range techniques, different analysis approaches and programs, and
prediction from satellite measurement and modeling of tides. Lunar laser ranges are
sensitive to tidal acceleration in mean anomaly as well as increasing semimajor axis.
Concerning the evidence for a lower recession rate offered by Riofrio’s lunar anomaly
paper [1]: (1) the geological record does not establish the present rate, (2) modeling of
one tidal component does not give the total lunar tidal acceleration and recession, and
(3) there is a geophysical origin for the nontidal acceleration of Earth rotation. The
cosmological suggestion that the speed of light is slowing has been tested and does not
match the prediction. The alleged lunar orbit anomaly does not exist and cosmological
inferences are not warranted. Tides evolve, but the speed of light remains steady.
Endnotes
a
The tidal acceleration in mean anomaly and mean longitude are nearly the same.
The tidal accelerations of the argument of perigee and node are smaller by two orders
of magnitude. The practice of giving the acceleration in mean longitude is a convention
from the days of optical observations of angles.
b
The computation of dissipation-induced orbit and rotation changes from each global
tidal component involves a ratio of Love number k2 divided by a specific dissipation Q
for the component. Large values of k2/Q are referred to as more effective here. For a
global representation of tides, although most of the phase-shifted tide comes from the
oceans, the in-phase part comes mainly from solid-body tides. For the global M2 tide,
LLR analysis has k2/Q ≈ 0.32/13 with a phase shift of 4.5˚.
c
The orbital tidal acceleration is mainly sensitive to dissipation from tides raised by
the Moon on the Earth, with a 1% effect from tides raised by the Earth on the Moon
[11,29]. However, the tidal deceleration of the Earth’s rotation is sensitive to tides raised
on the Earth by both Moon and Sun. Also, the K1 tidal dissipation affects rotation, but
not orbit. Consequently, although the larger part of the tidal deceleration of rotation is
from tidal components that also affect the orbit, a tidal model is required for part of
the rotation computation. This can be as simple as a constant phase shift, or as sophisticated as an ocean model such as [12,13].
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Abbreviation
JPL: Jet propulsion laboratory; LLR: Lunar laser ranging.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
JGW and DHB generated the lunar tidal acceleration solution. JGW, SGT, and DHB cooperated for the speed of light
solution. JGW drafted the manuscript. All authors read, edited, and approved the final manuscript.
Authors’ information
JGW provides theoretical developments, computational formulations, and oversees data solutions for LLR data analysis.
SGT participates in LLR studies concerning gravitational physics and cosmology. DHB performs the LLR data analysis
and maintains the LLR code.
Acknowledgment
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. Copyright 2013 California
Institute of Technology. Government sponsorship acknowledged.
Received: 17 July 2013 Accepted: 14 January 2014
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Cite this article as: Williams et al.: The past and present Earth-Moon system: the speed of light stays steady as
tides evolve. Planetary Science 2014 :2.
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