Based on the recent finding that the difference in proper time of two clocks
in prograde and retrograde equatorial orbits about the Earth is of the order
10^{-7}s per revolution, the possibility of detecting the terrestrial
gravitomagnetic field by means of clocks carried by satellites is discussed. A
mission taking advantage of this influence of the rotating Earth on the proper
time is outlined and the conceptual difficulties are briefly examined.Comment: Talk given at the 32nd COSPAR Scientific Assembly, held at Nagoya,
  Japan, 12-19 July 1998, 4 pages LATE

Gronwald, F.

Lichtenegger, H. I. M.

Mashhoon, B.

English

arXiv

http://arxiv.org/abs/gr-qc/9808017Based on the recent finding that the difference in proper time of two clocks in prograde and retrograde equatorial orbits about the Earth is of the order 10^{-7}s per revolution, the possibility of detecting the terrestrial gravitomagnetic field by means of clocks carried by satellites is discussed. A mission taking advantage of this influence of the rotating Earth on the proper time is outlined and the conceptual difficulties are briefly examined.Talk given at the 32nd COSPAR Scientific Assembly, held at Nagoya, Japan, 12-19 July 1998

Lichtenegger, Herbert

Mashhoon, Bahram

University of Missouri: MOspace

arXiv:gr-qc/9808017v1  5 Aug 1998ON DETECTING THE GRAVITOMAGNETIC FIELD OFTHE EARTH BY MEANS OF ORBITING CLOCKSH.I.M. Lichtenegger1, F. Gronwald2 and B. Mashhoon31Institut fu¨r Weltraumforschung, O¨sterreichische Akademie der Wissenschaften, A-8010 Graz, Austria,2Institut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, D-50923 Ko¨ln, Germany,3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USAABSTRACTBased on the recent finding that the difference in proper time of two clocks in prograde and retrogradeequatorial orbits about the Earth is of the order ∼ 10−7 s per revolution, the possibility of detecting theterrestrial gravitomagnetic field by means of clocks carried by satellites is discussed. A mission taking ad-vantage of this influence of the rotating Earth on the proper time is outlined and the conceptual difficultiesare briefly examined.INTRODUCTIONGeneral relativity was designed to be a field theory formally similar to electrodynamics, yet some basic dif-ferences remain. However, in the case of low velocities and weak fields, the formal analogy between gravityand electromagnetism gives rise to a number of similar phenomena, known as gravitoelectromagnetism. Dueto the weakness of the gravitomagnetic effects, however, their existence has yet to be verified.The proof whether rotating masses do indeed modify the gravitational field is important, both from a phys-ical as well as a philosophical point of view, since it is intimately related to our view of space as being eitherrelational or absolute. Hitherto discussed solely on a philosophical level, Newton raised the problem of thestatus of space on a physical basis by ascribing to absolute space a physical reality which becomes apparentvia inertial forces. By criticising Newton’s concept of abolute space, Mach stated that dynamics shouldnot contain absolute elements since in his positivistic view observable effects should have observable causes.Because inertial forces are observed only to occur in reference frames accelerated with respect to the fixedstars, for Mach it was therefore obvious to suppose that the distribution of matter in the universe deter-mines the local inertial frames and the reliance on absolute space becomes superfluous (Mach’s principle).Although it is still a matter of debate to what extent general relativity complies with Mach’s principle (seee.g. Mashhoon, 1988, and Barbour and Pfister, 1995), it is generally believed that certain gravitomagneticeffects are the most direct manifestations of Machian ideas in general relativity (Mashhoon, 1993). Hencethe study of these phenomena can help to clarify one of the most puzzling questions in physics: Where doesinertia come from?Several space missions have been proposed to measure gravitomagnetism directly, among them GravityProbe-B, LAGEOS III and the Superconducting Gravity Gradiometer Mission (see e.g. Ciufolini andWheeler, 1995). The Gravity Probe-B satellite will carry four superconducting spherical quartz gyroscopesto measure the precession of the gyroscopes relative to the distant stars. LAGEOS III could be launchedin an orbit whose inclination is supplementary to that of LAGEOS I and to measure the precession of theline of nodes of these two satellites. The Superconducting Gravity Gradiometer Mission envisages orbitingan array of three mutually orthogonal superconducting gravity gradiometers around the Earth to measuredirectly the contribution of the gravitomagnetic field to the tidal gravitational force.In contrast to the above proposals, in this paper we consider the possibility to detect gravitomagnetism bymeans of clocks, i.e. by measuring the difference in proper time displayed by two clocks orbiting in pro- andretrograde direction around the Earth.THE GRAVITOMAGNETIC CLOCK EFFECTThe gravitational field outside a rotating body of mass M and angular momentum J is given by the Kerrmetricds2 =(1−Rs rρ2)c2dt2 +2Rs rρ2a sin2 θ cdt dφ−ρ2∆dr2 − ρ2dθ2 − sin2 θ[r2 + a2 +Rs rρ2a2 sin2 θ]dφ2, (1)withρ2 = r2 + a2 cos2 θ, ∆ = r2 −Rsr + a2anda =JcM, Rs =2GMc2.Restricting ourselves to a circular equatorial orbit, one of the geodesic equations of Eq. (1) becomesdφdt≡ Ω = Ω0[±1−acΩ0 +O(a2c2)]∼= ±Ω0 +ΩLT, (2)where the gravitomagnetic correction in the equatorial plane to the Kepler period Ω0 = (GM/r3)12 (asdetermined by static observers at infinity) is just the Lense-Thirring precession ΩLT = −GJ/c2r3. Nowwe consider two clocks moving along pro- and retrogarde circular equatorial orbits, respectively, about theEarth. According to Eq. (2), the difference in proper time for the two clocks after some fixed coordinatetime t, say, one Kepler period t = T0 = 2pi/Ω0 becomes (τ+ and τ− denote the proper time along the pro-and retrograde orbits, respectively)(τ+ − τ−)t=T0∼= 12piGJc4r≈ 3× 10−16 s, (3)where we have inserted the values M = 6 × 1024 kg, J = 6 × 1033 kgm2 s−1 and r = 7000 km. This isa very tiny effect which, even apart from all perturbations, is not detectable with today’s atomic clocktechnology and which is probably the reason why clocks are barely considered as proper means to measurethe gravitomagnetic contribution of the Earth’s rotation. However, as recently pointed out by Cohen andMashhoon (1993), this time difference is considerably enlarged when calculating the difference in propertime of two counter-orbiting clocks for some fixed angular interval rather than for some fixed time interval.Integration over, say, 2pi yields (see Gronwald et al., 1997)(τ+ − τ−)φ=2pi ∼= 4piJMc2≈ 1× 10−7 s (4)and it is interesting to note that this difference is independent of both G and r. The relatively large valuein Eq. (4) could underlie an experiment to directly measure the terrestrial gravitomagnetic field (calledGravity Probe-C (GP-C) in Gronwald et al. (1997)). The above analysis can be extented to circular orbitswith finite inclination (Theiss, 1985), showing that the time difference decreases with increasing inclination;for a polar orbit the effect vanishes as it is expected because of symmetry.ERROR SOURCESAlthough the value in Eq. (4) is several orders of magnitude larger than that in Eq. (3) and well within themeasurement capabilities of modern atomic clocks ((τ+ − τ−)/T0 ∼ 10−11 for a near-Earth orbit), it is nota simple task to seperate the time difference induced by gravitomagnetism from all other effects resultinglikewise in a time shift between the clocks. The main error sources that may affect the experiment can bedivided into 4 categories and are listed below:1. Errors from orbital injection errors2. Errors from gravitational perturbations- Static odd zonal harmonic perturbations- Static non-zonal harmonic perturbations- Non-linear harmonic perturbations- Solid and ocean Earth tides- Sun, Moon and planetary tidal accelerations- Relativistic perturbations3. Errors from non-gravitational perturbations- Direct solar radiation pressure- Terrestrial radiation pressure (albedo, infrared radiation)- Satellite eclipses- Anisotropic thermal radiation (Yarkovsky effect)- Thermal thrust (Yarkovsky-Schach effect)- Poynting-Robertson effect- Atmospheric drag- Drag from interplanetary dust- Earth’s plasma environment (charged particle drag)- Earth’s magnetic field (magnetic despin)4. Errors from uncertainties in the determination of the orbital parametersAfter launch, the orbital elements have to be adjusted by inflight corrections as close as possible to thevalues desired, i.e. to zero eccentricity and inclination in order to maximize the effect. Further, an accuratetracking of the satellites is required since uncertainties in the radial and azimuthal location of the clocksreadily cover the gravitomagnetic effect. A simple error analysis yields ∆r/r ∼ ∆φ/φ ∼ Ω0J/Mc2 ∼ 10−11,i.e. one must keep track of the satellites with an accuracy of ∆r ≤ 0.1 mm and ∆φ ≤ 10−2 mas perrevolution for orbits with ∼ 7 000 km altitude (Gronwald et al., 1997). Equivalently, accelerations down to∆r/T 20 ∼ 10−11 g should be taken into account.Gravitational perturbations due to the nonsphericity of the Earth will at least partially cancel out for twosatellites following opposite orbits. While current models of the static and tidal Earth’s gravitational fieldinduce orbital errors in all three spatial directions of several mm (Ciufolini et al., 1998), the gravitationalinfluence of other celestial bodies on the satellites is essentialy due to the Moon (10−7 g) and the Sun(10−8 g) and can be modeled with sufficient accuracy (Gronwald et al., 1997). Finally, accelerations dueto relativistic effects other than the relativistic Kerr corrections to the terrestrial field can be neglected(Mashhoon and Theiss, 1982, Ashby and Bertotti, 1984).Several non-gravitational mechanisms which perturb the orbits of satellites have been modeled in some detailin the course of data analysis of laser-ranged satellites (e.g. Rubincam, 1982). The observed average dragon the LAGEOS satellite is ∼ 10−12 m/s2 and is due to Yarkovsky thermal, neutral and charged particledrag (Rubincam, 1990); thermal thrust induces a secular variation in inclination of LAGEOS of the order of∼ 1mas/yr (Farinella et al., 1990). Although the perturbations will depend on inclination and are also con-trolled by the eclipses (the LAGEOS satellites are on nearly polar orbits), non-gravitational perturbationson LAGEOS-type satellites (i.e. spherically symmetric, small ratio of cross-sectional area to mass satellites)along equatorial orbits are not expected to differ significantly from those in polar orbits. Based on theexperience of LAGEOS’s orbital variations, it might thus be possible to correct for the non-gravitationalperturbations to the desired order without the use of drag-free satellites; however, a conclusive answer can-not be given without a thorough analysis of these effects.SUMMARY AND CONCLUSIONBased on the fact that the difference in proper time of two counter-orbiting clocks along circular equatorialorbits after completing one revolution each is several orders of magnitude larger than the time differenceafter one Kepler period, a mission is proposed to directly measure the gravitomagnetic field of the Earthvia spaceborne atomic clocks. While many of the effects causing deviations from an ideal circular Keplerianorbit can be incorporated into the analysis, the accurate tracking of the satellites and the correct modeling ofthe dynamical part of the Earth gravity field seem to be the most difficult tasks for a Gravity Probe-C mis-sion. However, the reported clock effect is cumulative and after 100-1000 revolutions it should be detectableeven with today’s tracking techniques. Also, the next generation of static and time-dependent terrestrialgravity field models is expected to possess a precision sufficient for GP-C. Hence we consider GP-C as analternative way to detect gravitomagnetism, which ideally completes missions like Gravity Probe-B andappears technically less demanding.REFERENCESAshby, N. and B. Bertotti, Relativistic Perturbations of an Earth Satellite, Phys. Rev. Lett., 52, 485–488(1984).Barbour, J. and H. Pfister, editors, Mach’s Principle: From Newton’s Bucket to Quantum Gravity, volume 6of Einstein Studies, Birkha¨user (1995).Ciufolini, I. and J.A. Wheeler, Gravitation and Inertia, Princeton Series in Physics, Princeton (1995).Ciufolini, I., E. Pavlis, F. Chieppa, E. Fernandes-Vieira, and J. Perez-Mercader, Test of General Relativityand Measurment of the Lense-Thirring Effect with Two Earth Satellites, Science, 279, 2100–2103(1998).Cohen, J.M. and B. Mashhoon, Standard Clocks, Interferometry, and Gravitomagnetism, Phys. Lett. A,181, 353–358 (1993).Farinella, P., A.M. Nobili, F. Barlier, and F. Mignard, Effects of Thermal Thrust on the Node and Inclinationof Lageos Astron. Astrophys., 234, 546–554 (1990).Gronwald, F., E. Gruber, H. Lichtenegger, and R.A. Puntigam, Gravity Probe Clock - Probing the Gravit-omagnetic Field of the Earth by Means of a Clock Experiment, in Fundamental Physics in Space, pp.29–37, ESA SP-420 (1997).Mashhoon, B. and D.S. Theiss, Relativistic Tidal Forces and the Possibility of Measuring Them, Phys.Rev. Lett., 49, 1542–1545 (1982).Mashhoon, B., Complementarity of Absolute and Relative Motion, Phys. Lett. A, 126, 393–399 (1988).Mashhoon, B., On the Relativity of Rotation, in Directions in General Relativity, vol. 2, edited by B. L.Hu and T. A. Jacobson, pp. 182–194, Cambridge University Press, Cambridge (1993).Rubincam, D.P., On the Secular Decrease in the Semimajor Axis of LAGEOS’s Orbit, Celest.Mech., 26,361–382 (1982).Rubincam, D.P., Drag on the LAGEOS Satellite, J.Geophys. Res., 95, 4881–4886 (1990).Theiss, D.S., A General Relativistic Effect of a Rotating Spherical Mass and the Possibility of Measuring itin a Space Experiment, Phys. Lett. A, 109, 19–22 (1985).

On detecting the gravitomagnetic field of the earth by means of orbiting clocks

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