VOLUME 56, NUM@ERA PHYSICAL REVIEW LETTERS 27 JANUARY 1986

Measurement of the Lense-Thirring Drag on High-Altitude,Laser-Ranged Artificial Satellites

Ignazio CiufoliniCenter for Theoreticai Physics, Center for Relativity, and Physics Department, University of Texas, A ustin, Texas 787I2

(Received 16 October 1984; revised manuscript received 19 April 1985)

%e describe a new method of measuring the Lense- Thirring relativistic nodal drag usingLAGEOS together with another similar high-altitude, laser-ranged satellite with appropriatelychosen orbital parameters. %e propose, for this purpose, that a future satellite such as LAGEOS IIhave an inclination supplementary to that of LAGEOS. The experiment proposed here would pro-vide a method for experimental verification of the general relativistic formulation of Mach's princi-

ple and measurement of the gravitomagnetic field.

PACS numbers: 04.80.+z

In special and general relativity there are severalprecession phenomena associated with the angularmomentum vector of a body. If a test particle is orbit-ing a rotating central body, the plane of the orbit ofthe particle is dragged by the intrinsic angular momen-tum J of the central body, in agreement with the gen-eral relativistic formulation of Mach's principle. '

In the weak-field and slow-motion limit the nodallines are dragged in the sense of rotation, at a rategiven by

fI = [2/a'(I —e')'l'] J, (1)

where a is the semimajor axis of the orbit, e is the ec-centricity of the orbit, and geometrized units are used,i.e. , G = c = 1. This phenomenon is the Lense-Thirring effect, from the names of its discoverers in1918.2

In addition to this there are other precessionphenomena associated with the intrinsic angularmomentum or spin S of an orbiting particle. In theweak-field and slow-motion limit the vector Sprecesses at a rate given by' dS/dr = 0 x S where

0=———,vxa+ —,vx 7U+ ——J+1 3 1 3(J r)rf3 2

(2)

where v is the particle velocity, a—= dv/dr —'7U is itsnongravitational acceleration, r is its position vector, ris its proper time, and Uis the Newtonian potential.

The first term of this equation is the Thomas preces-sion. 3 It is a special relativistic effect due to the non-commutativity of nonaligned Lorentz transformations.It may also be viewed as a coupling between the parti-

cle velocity v and the nongravitational forces acting onit.

The second (de Sitter4-Fokkers) term is general re-lativistic, arising even for a nonrotating source, fromthe parallel transport of a direction defined by S; itmay be viewed as spin precession due to the couplingbetween the particle velocity v and the static—g p o

= 0 and gtp = 0—part of the space-timegeometry.

The third (Schiff ) term gives the general relativisticprecession of the particle spin S caused by the intrinsicangular momentum J of the central body —gtoe0.

We also mention the precession of the periapsis ofan orbiting test particle due to the angular momentumof the central body. This tiny shift of the perihelion ofMercury due to the rotation of the Sun was calculatedby de Sitter in 1916.7

All these effects are quite small for an artificial sa-tellite orbiting the Earth.

We propose here to measure the Lense-Thirringdragging by measuring the nodal precession of laser-ranged Earth satellites. We shall show that two satel-lites would be required; we propose that LAGEOSS 'o

together with a second satellite LAGEOS X with oppo-site inclination (i.e. , with I = 180 —I, where I= 109.94' is the orbital inclination of LAGEOS)would provide the needed accuracy.

The major part of the nodal precession of an Earthsatellite is a classical effect due to deviations fromspherical symmetry of the Earth's gravity field—quadrupole and higher mass moments. " These de-viations from sphericity are measured by the expan-sion of the potential U(r) in spherical harmonics.From this expansion of U(r) follows" the formula forthe classical precession of the nodal lines of an Earthsatellite:

R@0,(„S= ——fI0

'2 3 2cosI R@ 1+ 2e

2, ' J2+ J4 — (7 sin'I —4), +. . . ',1 e2 2 8 a 1 —e

(3)

278

VoLUME 56, NUMaER 4 PHYSICAL REVIEW LETTERS 27 JANUARY 1986

where 8 @ is the mean equatorial radius, the J2„arethe even zonal harmonic coefficients of the gravita-tional field, n —= 27r/P is the orbital mean motion, andP is the period.

Clearly the orbital parameters n, a, e, and I must bevery accurately known to identify the classical contri-bution to the precession. We investigate these pointsbelow„but the difficulty of measurement of theLense-Thirring dragging is apparent from the followingfacts: LAGEOS has a classical precession of —126'per year, while its Lense-Thirring precession is only 31arc msec/yr. Our principal difficulty arises from poorknowledge of certain moments of the potential of theEarth.

A way of overcoming this problem would be to mea-sure accurately J2,J4,J6, . . . , by orbiting severalhigh-altitude, laser-ranged satellites, plus LAGEOS tomeasure the Lense-Thirring effect. A better solutionwould be to orbit polar satellites. In 1976, Van Pattenand Everitt'2 proposed the measurement of theLense-Thirring effect with two counter-rotating drag-free satellites in polar orbit, i.e., counter-rotating inthe same plane perpendicular to the equator. Becausethe classical precession is proportional to cos(1),which vanishes for polar orbits with I = m/2, the con-tribution to the nodal precession due to the quadrupoleand higher moments of the Earth is zero for polar sa-tellites. However, in order to measure the inclinationwith the technology then available to sufficient accura-cy to extract the Lense-Thirring precession, it wouldhave been necessary to use two counter-rotating drag-free polar satellites, measuring their distance withsatellite-to-satellite Doppler ranging while the satelliteswere near the poles.

Today, however, the tracking accuracy of satelliteshas improved dramatically. The knowledge of the or-bits of some artificial satellites using ground-basedpulsed laser ranging systems will soon be sufficient tomeasure general relativistic perturbations. '3'4 Conse-quently, a general method'5' to measure the Lense-Thirring effect would be to have two laser-ranged sa-tellites with any arbitrary pair of opposite (i.e„ I and180' —I) inclinations to allow the separation of theclassical and the relativistic precessions.

In particular LAGEOS, launched in 1976, has thebest known orbit of any satellite because of its highaltitude —semimajor axis = 12 270 km —and its designto minimize air drag and radiation ' pressure. Forthis satellite the Lense-Thirring effect, Eq. (1), due tothe Earth's intrinsic angular momentum J = 5.9&& 104O gm cm2/sec, is ALT ——31 arc msec/yr, eastward.For an artificial satellite the measurable quantity isI1 —cu„where cu, is the Earth's time-dependent rota-tion rate. The estimates of UT1—in effect the Earth' srotation rate —with the use of lunar laser ranging(LLR) and very long-baseline interferometry (VLBI)

have an accuracy of fractions of a millisecond" of timecorresponding to an accuracy of a few arc millisecondsof rotation. Consequently, the nodal precession ismeasurable' with an accuracy of 1 or 2 arc msec/yr.Therefore, the relativistic precession is quite detect-able, if other contributions can be accurately account-ed for.

In addition to the Lense-Thirring effect it is alsonecessary to consider the Sun's de Sitter-Fokker ef-fect, i.e., the spin precession of the system "Earth plusan orbiting satellite" considered as a gyroscope, due tothe static field of the Sun. The average geodesic pre-cession projected into the Earth's equatorial plane is,for LAGEOS, '9 AnF = 17 arc msec/yr, eastward. Therelativistic node shift is dominated by the Lense-Thirring effect, and is in an eastward direction.

The orbital parameters n, a, and e appearing in Eq.(3) are measured with sufficient accuracy for our pur-pose via the LAGEOS laser ranging. However, theuncertainty in the inclination and thus in the classicalpart of the nodal precession may well be the dominantsource of uncertainty in determining the relativisticpart of the precession. Currently, a four-year span ofLAGEOS orbital inclination residuals (from a dynami-cally consistent reference orbit) when fitted with asimple model of periods 1050, 560, and 280 days leadsto residuals about this curve on the order of 2 arcmsec. '8 These residuals are determined from indepen-dent 15-d intervals of laser tracking data. The Lense-Thirring effect is a secular effect, while most classicalperturbations are of a periodic nature. Hence, it is theaverage (over a sufficiently long time) of the inclina-tion error that concerns us. Under visual inspectionthe residuals appear to be Gaussian distributed. In oneyear, there are 24 such data points, so that after oneyear one expects —2/424 —0.4 arc msec of randomerror in the estimate of I for the span of that year, im-

plying from Eq. (3) a contribution to the nodal preces-sion error of —3 arc msec/yr. This random errorpresumably can be reduced somewhat by an evenlonger period of data, though a careful study of sys-tematic errors and of the statistics will be required toestimate the ultimate accuracy. Secondly, there aresome correlations with our proposed second satellitewhich will lead to an improvement in the model, and areduction in the residuals. For example, the effects ofuncertainties in tidal parameters ean be reduced by ad-justing their values by use of data from the two satel-lites; changes in the inclination due to indeterminacyin the values of the J2„are equal and opposite for thetwo satellites, " thereby, from Eq. (3), correspondingto equal and opposite nodal precessions. Any changein the inclination due to polar motion is already solvedfor in the experimental value of the inclination 1(t),and errors introduced should be relatively shortperiod. A recent paper20 shows 2-arc-msec residuals

VQLUME 56, NUMBER 4 PHYSICAL REVIEW LETTERS 27 JANUARY 1986

between VLBI and LAGEOS polar motion data, basedon 5-d data intervals. This supports our expectation ofaverage pole accuracy over longer periods at the 0.5-arc-msec level or better in the near future, and thusrelatively small contributions to the uncertainty in theinclination.

%e propose an experiment that is easier and lesscostly to implement than polar satellites would be. ~epropose that the forthcoming2' satellite LAGEOS II(or some other future satellite LAGEOS X) be chosento have the following orbital parameters:

a LAGEOS X a LAGEOS —a

ILAoEos x = m —ILAoEos —+ I"

LAGEOS X ~LAGEOS —~

where a„ l„and e, are small. In other words, thesecond satellite must have almost the same semimajoraxis, period, and eccentricity as LAGEOS but an al-

most supplementary inclination. Because the classicalprecession (3) is linearly dependent on cos(I), withthis particular choice of orbital parameters the classicalnodal precessions of LAGEOS and LAGEOS X will benearly equal and opposite. On the other hand theLense-Thirring and the geodesic precession —whichare independent of I—will be the same for both satel-lites so that, apart from other perturbations, the sumof the two measured total precessions will be twice therelativistic precession of LAGEOS. The philosophy ofthis two-satellite proposal is that one uses external datato fix the even zonal harmonics of the Earth's poten-tial. One must conservatively take an external esti-mate of the error in these terms. Unfortunately, thepresent uncertainty 5J2„ in the numerical value of theeven zonal harmonic coefficients J2„ is of the or-der'8 22 5J2„= 10 6J2. Since the Lense-Thirring pre-cession A LT is 0LT = 6.9 x 10 A,i„„ the indeter-minacy in the theoretical value of the classical nodalprecession is more than 10 times the relativistic effectto be measured. The errors in these quantities limitthe acceptable differences a„ l„and e, between thetwo satellite orbits.

In order to have less than 3% individual contribu-tions to the experimental uncertainty (based on ex-pected improvements in the uncertainties 5J2„/J2 to—3x 10 7) the differences a„ l„and e, between theorbital parameters of LAGEOS and LAGEOS X mustbe less than a, = +16 km, I,= +0.13', and e,= +0.04. Moreover, in order to calculate the totalprecession O„„i it is also necessary to consider non-linear contributions from even zonal harmonics; con-tributions from nonzonal spherical harmonics; lunar,solar, and planetary gravitational perturbations; atmos-

~LAGEOS g LAGEOS + ~LAGEOS + ~LAGEOSexp class rel other (5)

where A,„~ is the experimentally measured value ofthe rate of nodal precession and Q„h„ is the small rateof advance, computable with sufficient accuracy, dueto classical disturbing forces other than those causedby the even zonal spherical harmonics.

pheric drag, radiation pressure; and earth tides. Allthese disturbing forces cause periodic and secular orbi-tal perturbations that are very small compared to per-turbations due to the Earth's quadrupole moment.The nonlinear contributions from the J2„coefficientsare equal and opposite for LAGEOS and LAGEOS X'and anyway, since the precession associated with (J2)is of the order 0

iJ i2= 10 Q,i„„ these contribu-

tions could be calculated with sufficient accuracy to al-low the determination of the Lense-Thirring effect.Similarly the secular nodal precessions associated withthe nonzonal spherical harmonics and lunar, solar, andplanetary perturbations can be calculated with suffi-cient accuracy to allow their extraction. '8 Concerningthe nodal precession associated with radiation pressureand earth tides, information on these perturbations isextracted from the time variation of 0 when mea-sured over a long enough period of time. '8 23 BecauseLAGEOS is a spherical satellite and has a very smallorbital eccentricity, the error in the calculated secularrate of nodal precession due to direct solar radiationpressure is small' compared to ALT. Regarding thedirect solar radiation pressure, the main effect for anearly symmetric satellite is due to eclipses of part ofthe orbit. The eclipse seasons for LAGEOS lastroughly 90 days and are separated by periods abouttwice that long. When the Sun is at equal distancesfrom the satellite orbital plane, but in opposite regions,that is, before and after the Sun crosses the orbitalplane, the out-of-plane components of the solar radia-tion pressure force are nearly equal and opposite; thecontributions of the satellite eclipse to the nodal pre-cession are nearly equal and opposite. For this reasonthe error, over one year, in the calculated secular rateof nodal precession due to the satellite eclipses issmall' compared to ALT. Systematic effects due tothe Earth's albedo and to small differences in the sa-tellite albedo for changes in the direction of the solarradiation with respect to the satellite rotation axis haveto be considered, particularly in terms of interactionwith eclipse-season effects. However, the prospectsfor modeling the radiation pressure and the atmos-pheric drag effects on the node with small errors aregood. ' A detailed analysis of various perturbingforces will be the subject of a following paper. '9

In summary we have

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Ver UME56, NCIMaER4 PHYSICAL REVIEW LETTERS 27 JANUARY 1986

For the second satellite, we have

+LAGFOS x LAGEos+ ~ ~~class class class~ (6)

=3~ &eieesCOSI

a (1 —e')'Qq 4ee,' —I, tanI + —

2J2+. . . ,

2 a 1 —e(6)

and the total nodal precession is

IILAGEos x I') LAGEos x+ I-'I LAGEos x+ 1'I LAGEos x 1'ILAGEos+ g II + I'ILAGEos+ j)LAGEos xexp class rel other class class rel other

Finally, from (5) and (7), we have

I'ILAGEos ~ LAGEos+ I'I LAGEos x II LAGEos I'ILAGEos x Q QII rel e e exp + exp other other class

(7)

In conclusion we stress that because of the increasedtracking accuracy of the orbits of artificial satellites us-ing laser ranging systems, the Lense-Thirring and deSitter-Fokker relativistic nodal precessions are nowsignificant and should be taken into account in thetheoretical determination of orbital perturbations ofhigh-altitude satellites such as LAGEOS. These rela-tivistic nodal precessions should also be taken into ac-count in geodesy calculations for a more accuratedetermination of the irregular gravitational field of theEarth, i.e., quadrupole and higher mass moments.

The experiment described above will provide an al-ternative verification, together with the Everitt-Fairbank experiment, 24 of Mach's principle and will

give at last a measurement of the Earth's gyrogravita-tional or magnetogravitational field. Finally, we ob-serve that orbiting a satellite with an inclination oppo-site to that of LAGEOS will also give a better deter-mination of UTI. 's'9 For these reasons I propose thatthe forthcoming high-altitude laser-ranged satelliteLAGEOS II or a similar future satellite have an in-clination Iof 70.06'.

I thank Professor P. Bender, Professor B. Bertotti,Professor M. Demianski, Professor B. DeWitt, Profes-sor L. Shepley, and Professor S. Weinberg for helpfuldiscussions and particularly Professor R. Matzner, Pro-fessor B. E. Schutz, Professor B. D. Tapley, ProfessorJ. A. Wheeler, and Professor R. J. Eanes. This workwas supported in part by National Science FoundationGrants No. PHY82-05717 and No. PHY8404931.

'C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravita-tion (Freeman, San Francisco, 1973), p. 1066.

2J. Lense and H. Thirring, Phys. Z 19, 156 (1918); En-glish translation by B. Mashhoon et al. , Gen. Relativ. Gravit.16, 711 (1984).

3L. H. Thomas, Philos. Mag. 3, I (1927).4W, de Sitter, Mon. Not. Roy. Astron. Soc. 77, 155

(1916).

SA. D. Fokker, Proc. K. Ned. Akad. Wet. 23, 729 (1920).6L. 1. Schiff, Proc. Natl. Acad. Sci. U.S.A. 46, 871 (1960).

de Sitter, Mon. Not. Roy. Astron. Soc. 76, 699(1916).

SD. E. Smith and P. J. Dunn, Geophys. Res. Lett. 7, 437(1980).

9C. F. Yoder etal , Nature . (London) 303, 757 (1983).'OR. J. Eanes, B. E. Schutz, and B. D. Tapley, to be pub-

lished.t&W. M. Kaula, Theory of Satellite 'Geodesy (Blaisdell, Walt-

ham, 1966).&2R. A. VanPatten and C. W. F. Everitt, Phys. Rev. Lett.

36, 629 (1976).~3For a discussion of the measurement of the perigee pre-

cession of LAGEOS, see N. Ashby and B. Bertotti, Phys.Rev. Lett. 52, 485 (1984).

&4A theoretical method to measure the Riemann curvaturetensor ~ith a generalized version of the geodesic deviationequation [I. Ciufolini, to be published] and using four laser-ranged test particles is described in I. Ciufolini andM. Demianski, to be published,

I. Ciufolini, dissertation, University of Texas at Austin,1984 (unpublished).

'6I. Ciufolini, Bull. Am. Phys. Soc. 6, 1169 (1985).~ %. E. Carter„D. S. Robertson, J. E. Pettey, B. D. Tapley,

B. E. Schutz, R. J. Eanes, and Miao Lufeng, Science(Washington, D.C.) 224, 957 (1984).

~88. D. Tapley, B. E. Schutz, and R. J. Eanes, private com-munication.

'9I. Ciufolini, to be published.2 D. S. Robertson, %. E. Carter, B. D. Tapley, B. E.

Schutz, and R. J. Eanes, Science (Washington, D.C.) 229,1259 (1985).

2lNational Aeronautics and Space Administration/PianoSpaziale Nazionale LAGEOS II Study Group, Final Report,1983 (unpublished).

22F. J. Lerch et a/. , J. Geophys. Res. A 84, 3879 (1979).2 The experiment should be done over a period of a few

years.24The experiment to measure the Schiff precession of the

spin axis of a gyroscope orbiting the Earth is described byC. %. F. Everitt, in Experimental Gravitation, edited by B.Bertotti (Academic, New York, 1973), p. 331.