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Measurements of Newton’s gravitational constant and the length

of day

J. D. Anderson1(a) , G. Schubert2, V. Trimble3and M. R. Feldman4

1Jet Propulsion Laboratory, California Institute of Technology - Pasadena, CA 91109, USA

2Department of Earth, Planetary and Space Sciences, University of California, Los Angeles

Los Angeles, CA 90095, USA

3Department of Physics and Astronomy, University of California Irvine - Irvine CA 92697, USA

4Private researcher - Los Angeles, CA 90046, USA

PACS 04.80.-y – Experimental studies of gravity

PACS 06.30.Gv – Velocity, acceleration, and rotation

PACS 96.60.Q- – Solar activity

Abstract –About a dozen measurements of Newton’s gravitational constant, G, since 1962 have

yielded values that diﬀer by far more than their reported random plus systematic errors. We

ﬁnd that these values for Gare oscillatory in nature, with a period of P= 5.899 ±0.062 yr, an

amplitude of (1.619 ±0.103) ×10−14 m3kg−1s−2, and mean-value crossings in 1994 and 1997.

However, we do not suggest that Gis actually varying by this much, this quickly, but instead that

something in the measurement process varies. Of other recently reported results, to the best of

our knowledge, the only measurement with the same period and phase is the Length of Day (LOD

- deﬁned as a frequency measurement such that a positive increase in LOD values means slower

Earth rotation rates and therefore longer days). The aforementioned period is also about half of

a solar activity cycle, but the correlation is far less convincing. The 5.9 year periodic signal in

LOD has previously been interpreted as due to ﬂuid core motions and inner-core coupling. We

report the G/LOD correlation, whose statistical signiﬁcance is 0.99764 assuming no diﬀerence in

phase, without claiming to have any satisfactory explanation for it. Least unlikely, perhaps, are

currents in the Earth’s ﬂuid core that change both its moment of inertia (aﬀecting LOD) and

the circumstances in which the Earth-based experiments measure G. In this case, there might be

correlations with terrestrial magnetic ﬁeld measurements.

Introduction. – Newton’s gravitational constant, G,1

is one of a handful of universal constants that comprise our2

understanding of fundamental physical processes [1] and3

plays an essential role in our understanding of gravitation,4

whether previously in Newton’s attractive gravitational5

force between two massive bodies m1, m2of magnitude [2]6

F=Gm1m2

r2,(1)

where ris their separation distance, or currently as7

the proportionality constant in the interaction between8

energy-momentum content Tab (the stress-energy tensor)9

and space-time curvature Gab (Einstein tensor) in Ein-10

(a)Retired.

stein’s general relativity [3,4] 11

Gab =Rab −1

2gabR= 8πGTab,(2)

in units where the local speed of light in vacuum c= 1. 12

Yet, experimental determination of Newton’s gravitational 13

constant remains a challenging endeavor. As reviewed in 14

[5], several measurements over the last thirty years appear 15

to give inconsistent values for G, of course an issue for our 16

understanding of this universal constant. Our purpose 17

with this letter is to inform the reader of a one-to-one 18

correlation between an apparent temporal periodicity in 19

measurements of G, generally thought to result from in- 20

consistency in measurements, with recently reported oscil- 21

latory variations in measurements of LOD [6]. LOD refers 22

to the excess of the duration of the day (observed period 23

p-1

J. D. Anderson et al.

1985

1990

1995

2000

2005

2010

2015

6.671

6.672

6.673

6.674

6.675

6.676

Date HyrL

GH10-11 m3s-2kg-1L

NIST-82

TR&D-96

LANL-97

HUST-05

UWash-00

BIPM-01

UWup-02

MSL-03

JILA-10

UZur-06

HUST-09

BIPM-13

LENS-14

Fig. 1: Result of the comparison of the CODATA set of G

measurements with a ﬁtted sine wave (solid curve) and the 5.9

year oscillation in LOD daily measurements (dashed curve),

scaled in amplitude to match the ﬁtted Gsine wave. The

acronyms for the measurements follow the convention used by

CODATA, with the inclusion of a relatively new BIPM re-

sult from Quinn et al. [11] and another measurement LENS-14

from the MAGIA collaboration [18] that uses a new technique

of laser-cooled atoms and quantum interferometry, rather than

the macroscopic masses of all the other experiments. The green

ﬁlled circle represents the weighted mean of the included mea-

surements, along with its one-sigma error bar, determined by

minimizing the L1 norm for all 13 points and taking into ac-

count the periodic variation.

of rotation of the Earth) relative to a standard unit and24

is calculated by taking the diﬀerence between atomic time25

(TAI) and universal time (UT1) divided by the aforemen-26

tioned standard unit of 86400 SI s [7]. Variations in LOD27

can be used to determine changes in the Earth’s rotation28

rate eﬀectively providing a means to examine geophysical29

and atmospheric processes [8].30

For the following discussion, we emphasize that our G31

analysis and LOD analysis (a veriﬁcation of the proce-32

dures employed in [6]) are very much independent of one33

another with the determined ﬁtting parameters for both34

the period and phase of the periodicities in these measure-35

ments coinciding in near perfect agreement. Although we36

recognize that the one-to-one correlation between the ﬁt37

to the Gmeasurements and the LOD periodicity of 5.938

years could be fortuitous, we think this is unlikely, given39

the striking agreement shown in Fig. 1. Furthermore, after40

taking into account this ﬁtted oscillatory trend in the G41

measurements, we obtain agreement amongst the diﬀerent42

experiments mentioned in [5] with a weighted mean value43

for Gof (6.673899 ±0.000069) ×10−11 m3kg−1s−2.44

Methods. – In the July 2014 issue of Physics To-45

day, Speake and Quinn [5] lay out the problem and review46

the history of seemingly inconsistent measurements of the47

gravitational constant G. They plot twelve Gdetermina-48

tions, along with one-sigma error bars, extending from an49

experiment by Luther and Towler at the National Bureau50

of Standards (NBS) in 1982 [9] to their own at BIPM in51

2001 and 2007 (the latter of which was published in 2013) 52

[10, 11], two measurements in good agreement with each 53

other, but not with the other 10 measurements. Though 54

the vertical scale of years when the measurements were 55

made is not linear, there is a striking appearance of a pe- 56

riodicity running through these values, characterized by 57

a linear drift which suddenly reverses direction and then 58

repeats more than once. 59

With this pattern in mind, we compute a periodogram 60

for the measured Gvalues versus estimated dates of when 61

the experiments were run. A single clear period of 5.9 62

years emerges. The data for our Ganalysis were ob- 63

tained directly from Table XVII in the 2010 CODATA 64

report published in 2012 [1]. There are 11 classical mea- 65

surements made at the macroscopic level. To those we 66

added two more recent data points, another macroscopic 67

measurement, which we label BIPM-13, and the ﬁrst ever 68

quantum measurement with cold atoms, labeled LENS-14. 69

Next we used our best estimates of when the experiments 70

were run, not the publication dates, for purposes of gen- 71

erating a measured Gvalue versus date data ﬁle, with 72

one-sigma errors included too. These dates were obtained 73

from the respective articles. This gives us the best data 74

set possible, deﬁned by the measured Gvalues used for 75

the CODATA recommendation plus two more published 76

after 2012. 77

We ﬁt with the raw standard errors, σi, provided with 78

each of the Gmeasurements and used a numerical mini- 79

mization of the L1 and L2 norms of the weighted residu- 80

als, ri/σi, where the residuals are about a ﬁtting model 81

of a single sine wave, a0+a1cos ωt +b1sin ωt, four pa- 82

rameters in all with 13 measurements. Results for the ﬁt 83

to the 13 measured Gvalues are summarized in Fig. 1. 84

The L2 minimization is equivalent to a weighted least 85

squares ﬁt, yet the L1 minimization (solid line in Fig. 1) 86

is a more robust estimator that discriminates against out- 87

liers. Both yield excellent ﬁts with a suggestion that two 88

measurements at Moscow [12] and from the MAGIA col- 89

laboration [18] are outliers. However, the Moscow value is 90

known to suﬀer from an unexplained temporal drift [12] 91

and the cold-atom value could be fundamentally diﬀerent 92

(Gat the quantum level). Still, we refrain from specu- 93

lating further on the cold-atom outlier until more micro- 94

scopic measurements of Gare obtained by diﬀerent exper- 95

imental groups. The other 11 measurements are consis- 96

tent with the L1 ﬁtting curve at the one-sigma level or 97

better. Figure 1 appears to provide convincing evidence 98

that there exists a 5.9 year periodicity in the macroscopic 99

determinations of Gin the laboratory with variations at 100

the level of ∆G/G ∼2.4×10−4about a mean value of 101

6.673899 ×10−11 m3kg−1s−2, close to the value recom- 102

mended by CODATA in 2010 [1] but with a much smaller 103

standard error of 10.3 ppm instead of the CODATA rec- 104

ommended error of 120 ppm. 105

The most accurate determination by the Washington 106

group [13] with a standard error of 14 ppm now falls 107

squarely on the ﬁtting curve. Because the two BIPM mea- 108

p-2

Measurements of Newton’s gravitational constant and the length of day

surements were made at the peak of the ﬁtting curve, they109

now not only agree, but they are consistent with all other110

measurements. Notably, the measurement with a simple111

pendulum gravity gradiometer at JILA is no longer bi-112

ased to an unacceptably small value, but like the BIPM113

measurements it falls right on the ﬁtting curve, but at the114

minimum of the sine wave. The Huazhong measurement115

is also at the minimum of the curve.116

Results. – With the 5.9 year periodicity in the G117

measurements accepted, the question arises as to what118

could be the cause and what does it mean. The only thing119

we can think of is a correlation with a 5.9 year periodicity120

in the Earth’s LOD, published by Holme and de Viron last121

year [6]. The International Earth Rotation and Reference122

Systems Service (IERS), established in 1987, maintains123

downloadable data ﬁles containing daily values of sev-124

eral parameters related to Earth orientation and rotation.125

The ﬁles extend from 1962 January 01, when the Consul-126

tative Committee on International Radio (CCIR) estab-127

lished Universal Time Coordinated (UTC) as the standard128

for time keeping, to the most current date available. We129

extract two rotation ﬁles, the ﬁrst is the diﬀerence UT1-130

UTC in seconds and the second the LOD, also expressed131

in seconds, along with daily estimates of standard errors132

for both. There is also a piecewise constant ﬁle in integer133

seconds for the standard of atomic time TAI minus UTC.134

By diﬀerencing these two ﬁles the phase of the Earth ro-135

tation is obtained as measured against a uniform atomic136

time. This diﬀerence can be thought of as a continuous137

phase function φ(t) in radians sampled once per day at the138

beginning of the day. It can be expressed in SI seconds, the139

units on the IERS ﬁles, by multiplying by the conversion140

factor 86400/2π. It essentially provides the time gained or141

lost over the years by a poor mechanical clock, the Earth,142

which runs slow with a loss of about 33 s over the 52 years143

of the downloaded ﬁle. Because of its name and units of144

seconds only, the second ﬁle LOD is more diﬃcult to in-145

terpret. It is also the gain or loss of time by the Earth,146

but only over the current day, and because of deﬁnitions147

there is a reversal in sign. When expressed as a contin-148

uous function of the Earth’s rotational frequency ν(t), it149

is simply ν0−˙

φ/2π, where ν0is an adopted frequency of150

rotation with sidereal period of 86164.098903697 s. The151

quantity ˙

φ/(2πν0) is small and can be taken to the ﬁrst152

order in all calculations.153

Formally, the spectral density of frequency is re-154

lated to the spectral density of phase by SLOD(f) =155

(2πf )2SUT1 (f), where fis the Fourier frequency. How-156

ever, a separate computation of the spectrum for each ﬁle157

shows that before 1994 either ﬁle can be used for analy-158

sis, but after the introduction of Global Positioning (GPS)159

data in 1993, the LOD data become more accurate by a160

factor of seven or more. This conclusion is consistent with161

the standard errors included with the data ﬁles of LOD162

and UT1-UTC. We show our estimate of the spectral den-163

sity for the LOD data in Fig. 2, obtained by weighted least164

-2.0 -1.5 -1.0 -0.5 0.0

-4

-3

-2

-1

0

1

Log f (yr -1 )

Log LOD2(ms2d-2 )

UT1 -TAI

Fig. 2: One-sided power spectral density per unit frequency for

LOD data over the years 1962 to 2014. The white-noise ﬂoor

is indicated by the horizontal solid line and corresponds to a

standard deviation of 0.54 ms d−1, achieved by introduction

of GPS data in 1993 and consistent with the daily estimates

of standard error archived with the LOD data. The upper

dashed curve corresponds to mean spectral density for the nu-

merical time derivative of the UT1 data, dependent on VLBI

data from radio sources on the sky. For the low end of the

spectrum the LOD and UT1 data both indicate a f−2random

walk, which with only 52 years of data can be confused with a

drift in the Earth’s rotation. At the high end, the underlying

spectrum indicates white LOD noise, but with a rich spectrum

from tidal torques and atmospheric loading at higher frequen-

cies not plotted. Although there is power in the region, there

is no suggestion of a single spectral line from the 5.9 year oscil-

lation, a term which must be extracted by analysis in the time

domain [6].

squares and SVD, but this time with 850 Fourier coeﬃ- 165

cients, 430 degrees of freedom, and 19169 observations. 166

The spectral resolution is 0.019 yr−1, which we oversam- 167

ple by a factor of four, and the frequency cut oﬀ is 2 yr−1,168

far short of the Nyquist frequency of 0.5 d−1. A window 169

function is not applied to the data. It introduces unde- 170

sirable artifacts into the low-frequency noise spectrum of 171

interest and does little to isolate spectral lines. The Gaus- 172

sian window produces a hint of a line at 5.9 yr, but only 173

a hint. We proceed to an analysis of the data in the time 174

domain. 175

The 5.9 year periodicity in the LOD data is plotted by 176

Holme and de Viron in Figure 2 of their paper [6]. Their 177

plot looks in phase with the ﬁt to the 13 Gvalues, but 178

in order to obtain an independent check on the reality 179

of the signal and for purposes of having a numerical sine 180

wave extending into 2014, we ﬁrst smooth the LOD data 181

with a Gaussian ﬁlter with a radius of 600 days and a stan- 182

dard deviation of 200 days. As a result, the high-frequency 183

noise at a period of one year and shorter is practically elim- 184

inated, and with little eﬀect on the low-frequency noise 185

spectrum. Next we ﬁt a cubic spline to the smoothed 186

data with a selection of knots or segments for the cubic 187

polynomials done by eye, such that the ﬁtting curve is suf- 188

ﬁciently smooth but with a negligible eﬀect on the 5.9 year 189

p-3

J. D. Anderson et al.

1985

1990

1995

2000

2005

2010

2015

6.671

6.672

6.673

6.674

6.675

6.676

Date HyrL

GH10-11 m3s-2kg-1L

NIST-82

TR&D-96

LANL-97

HUST-05

UWash-00

BIPM-01

UWup-02

MSL-03

JILA-10

UZur-06

HUST-09

BIPM-13

LENS-14

Fig. 3: Result of the comparison of our Gdata set with

the monthly mean of the total sunspot number, appropriately

scaled. The black curves represent solar activity as reﬂected in

the international sunspot number.

periodicity. The resulting LOD residuals are ﬁt with a sine190

wave of ﬁxed 5.9 year period which is then subtracted from191

the smoothed data. The same procedure is applied to the192

new smoothed data and the procedure repeated four times193

with the knots for the spline at closer spacing with each194

iteration. The ﬁnal result is the pure sine wave plotted195

as a dashed curve in Fig. 1. It agrees with the periodic196

signal found by Holme and de Viron. A removal of the ﬁt-197

ted spline representation of the random walk, and also the198

sine wave, from the smoothed data is all that is needed in199

order to reduce the LOD residuals about the ﬁt to a one-200

sigma noise level of 4.8µs d−1. The amplitude of the ﬁtted201

periodic signal is 92.64 ±0.18 µs d−1, reduced from the202

amplitude of 150 µs d−1[6] by the Gaussian smoothing,203

but with a well-determined period of 5.90076 ±0.00074 yr.204

With 99% conﬁdence the period lies between 5.898 and205

5.903 yr. The phasing of the sine wave is as shown in206

Fig. 1 with a standard error of 0.25 yr.207

The correlation between LOD and Gmeasurements in208

Fig. 1 is most likely of terrestrial origin, but the period of209

5.9 years is also close to one-half the principal period of210

solar activity. References [14] and [15] discuss in greater211

detail that a possible correlation between solar activity212

and LOD measurements is not unexpected. Solar activity213

has an eﬀect on mass distribution in the atmosphere which214

ultimately aﬀects the Earth’s axial moment of inertia. It215

is feasible that this eﬀect occurs at longer periods in the216

5.9-year range, as well as at much shorter periods, on the217

order of days, for which models exist [6].218

Consequently, we plot in Fig. 3 the monthly mean of219

the total sunspot number and also a 13-month smoothing220

curve, both shown in black. The two curves, again scaled221

to the magnitude of the Gdata, are taken directly from222

freely available downloads of data archived at www.sidc.be223

by WDC-SILSO, Royal Observatory of Belgium, Brussels.224

The smoothing is done by a standard tapered-boxcar ap-225

proach and is generally regarded as a good measure of226

solar activity. Although the Gmeasurements show a gen-227

eral agreement with solar cycle 23, which peaked around 228

2002, the long and unexpected minimum that followed, 229

and lasted until about 2010, is at odds with the rise in 230

Gvalues during that minimum. There is also a negative 231

correlation between the measurement from 1982 at the 232

National Bureau of Standards, labeled NIST-82, and the 233

sunspot number. It seems that solar activity can be dis- 234

regarded as a cause of the variations in Gmeasurements. 235

Conclusions. – Over the relatively short time span 236

of 34 years considered here, variations in the rotation of 237

the Earth can be considered either a random walk or pos- 238

sibly a drift. Over much longer time scales the rotation 239

must be slowing because of the transfer of spin angular 240

momentum to orbital angular momentum caused by tidal 241

friction of the Moon. Similarly, a real increase in Gshould 242

pull the Earth into a tighter ball with an increase in an- 243

gular velocity and a shorter day due to conservation of 244

angular momentum, contrary to the correlation shown in 245

Fig. 1. Thus, we do not expect that this behavior neces- 246

sarily points to a real variation in Gbut instead to some 247

yet-to-be determined mechanism aﬀecting both measure- 248

ments in a similar manner. 249

Importantly, if the observed eﬀect is connected with a 250

centrifugal force acting on the experimental apparatus, 251

changes in LOD are too small by a factor of about 105252

to explain the changes in Gfor the following reason. The 253

Earth’s angular velocity ωEis by deﬁnition 254

ωE=ω0(1 −LOD),(3)

where ω0is an adopted sidereal frequency equal to 255

72921151.467064 picoradians per second and the LOD is in 256

ms d−1(www.iers.org). The total centrifugal acceleration 257

is given by 258

ac=rsω2

01−2Asin 2π

P(t−t0),(4)

where Ais the amplitude 0.000150/86400 of the 5.9 year 259

sinusoidal LOD variation and rsis the distance of the ap- 260

paratus from the Earth’s spin axis. The maximum per- 261

centage variation of the LOD term is 3.47 ×10−9of the 262

steady-state acceleration, while ∆G/G is 2.4×10−4, hence 263

even the full eﬀect of the acceleration with no experimen- 264

tal compensation changes Gby only 10−5of the amplitude 265

in Fig. 1. Perhaps instead, the eﬀect is connected with 266

changing torques on the Earth’s mantle due to changing 267

motions in the core. Changes of circulation in the core 268

must be accompanied by changes in density variations in 269

the core causing variations in the gravitational accelera- 270

tion gin the laboratory. At least this mechanism links 271

both LOD and gravitational changes to changes in the 272

core although we do not immediately see how either of 273

these mechanisms could aﬀect measurements of Gin the 274

laboratory given the torsion balance schemes employed. 275

The least likely explanation is a new-physics eﬀect that 276

could make a diﬀerence in the macroscopic and micro- 277

scopic determinations of G. Perhaps a repetition of the 278

p-4

Measurements of Newton’s gravitational constant and the length of day

single 2014 quantum measurement over the next decade279

or so can show consistency with a constant value, al-280

though if the variations in Gmeasurements are caused281

by an unknown inertial or frame eﬀect, not by systematic282

experimental error, it likely applies at both the macro-283

scopic and the microscopic levels. The gravitational pa-284

rameter for the Sun, GM, is known to ten signiﬁ-285

cant ﬁgures from orbital motions in the Solar System286

(ssd.jpl.nasa.gov/?constants). The universal constant G287

does not vary at that scale, although Krasinsky and Brum-288

berg [16, 17] report a detection of an unexplained secular289

increase in the astronomical unit (AU) over the years 1976290

to 2008, which can be interpreted as an increase in GM

291

proportional to the cube of the AU. However the eﬀect on292

G, if real, is at the level of an increase of 3 parts in 1012

293

per year and undetectable with laboratory measurements294

of G. Nevertheless, the increase in GMis not explainable295

as an increase of the solar mass by accretion as opposed to296

the mass radiated away by solar luminosity [17]. Appar-297

ently, there does seem to be a secular or very long period298

(greater than 20000 years) Gvariation in the Solar Sys-299

tem, but of order 10−6smaller than the variation shown300

in Fig. 1.301

∗∗∗

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