http://arxiv.org/abs/gr-qc/0101089As a consequence of gravitomagnetism, which is a fundamental weak-field prediction of general relativity and ubiquitous in gravitational phenomena, clocks show a difference in their proper periods when moving along identical orbits in opposite directions about a spinning mass. This time shift is induced by the rotation of the source and may be used to verify the existence of the terrestrial gravitomagnetic field by means of orbiting clocks. A possible mission scenario is outlined with emphasis given to some of the major difficulties which inevitably arise in connection with such a venture

Hausleitner, W.

Lichtenegger, Herbert

Mashhoon, Bahram

University of Missouri: MOspace

arXiv:gr-qc/0101089v1  23 Jan 2001SOME ASPECTS ON THE OBSERVATION OF THEGRAVITOMAGNETIC CLOCK EFFECTH. Lichtenegger1, W. Hausleitner1, F. Gronwald2 and B. Mashhoon31Institut fu¨r Weltraumforschung, O¨sterreichische Akademie der Wissenschaften, A-8042 Graz, Austria,2Institut GET, Universita¨t Magdeburg, D-39106 Magdeburg, Germany,3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USAABSTRACTAs a consequence of gravitomagnetism, which is a fundamental weak-field prediction of general relativityand ubiquitous in gravitational phenomena, clocks show a difference in their proper periods when movingalong identical orbits in opposite directions about a spinning mass. This time shift is induced by the rotationof the source and may be used to verify the existence of the terrestrial gravitomagnetic field by means oforbiting clocks. A possible mission scenario is outlined with emphasis given to some of the major difficultieswhich inevitably arise in connection with such a venture.INTRODUCTIONAmong the essential predictions of General Relativity, gravitomagnetism (and gravitational waves) still lackdirect observational evidence. The only direct reference to date for the existence of gravitomagnetism isbased on the orbital data analysis of Lageos I and II, which suggests that the orbital precession due tothe spinning Earth corresponds with the Einsteinian prediction within ∼ 20% (Ciufolini et al., 1998). Thefirst space experiment directly designed to detect the gravitomagnetic field of the Earth will be GravityProbe B, which shall measure the Lense-Thirring precession of a couple of test gyros carried by a spacecraftalong a near-Earth polar orbit. In the following we describe an alternative space experiment to observe theterrestrial gravitomagnetic field, based on the temporal structure induced by the rotating Earth, and discussthe influence of the static part of the terrestrial gravitational field on the outcome of such a measurement.THE CLOCK EFFECTIn the weak field and low velocity limit, the field equations of General Relativity reduce to a set of Maxwell-type equations, thereby giving rise in this approximation to a number of phenomena similar to those known inelectrodynamics, among them the well-known Lense-Thirring precession, which is the gravitational analogueof the precession of a spinning dipole in a magnetic field. However, the gravitomagnetic field not only makesa test gyro to precess about the field lines but also affects the motion of a test body resulting in a difference inthe proper period for co- and counter-revolving particles. Again, this can be easily understood by comparisonwith electrodynamics: Consider a test charge moving along a circular orbit about a central charge. When aweak and homogeneous magnetic field with orientation perpendicular to the orbital plane is switched on, thetest charge experiences an additional Lorentz force, either adding to or opposing the Coulomb attraction,depending on the direction of the motion of the particle and the magnetic field, respectively. The particlecan, however, still circle along the same path as without the field, but now with a somewhat increased ordecreased velocity; exactly the same is expected to happen in the gravitomagnetic case. The reason for thisis the formal equivalence between the Lorentz force and the Coriolis force, i.e. Larmor’s theorem and itsgravitational analogue (Mashhoon, 1993). It follows that among two particles along identical but oppositeorbits about a rotating mass, one will move faster than the other, resulting in a difference of their revolutiontimes. Numerically, for circular equatorial orbits this time difference (called gravitomagnetic clock effect)to leading order is found to be (Cohen and Mashhoon, 1993, Gronwald et al., 1997)∆τ ≃ 4pi JMc2≈ 2× 10−7s, (1)where J and M are the angular momentum and mass of the central body, respectively, and the numericalvalue results upon inserting the appropriate values for the Earth. In comparison, for Sun and Jupiter,which have the largest specific angular momenta among all solar system bodies, the clock effect amounts to∆τSun ≃ 7× 10−5 s and ∆τJupiter ≃ 5× 10−5 s, respectively.Within the PPN formalism, for semiconservative theories and ignoring the Whitehead term and any pre-ferred location effects, an additional factor of (1 + γ + α1/4)/2 appears in Eq. (1) (Mashhoon et al., 2000),showing that the clock effect depends, as expected, on the same PPN parameter combination as the gravito-magnetic precession of a gyroscope (e.g. Will, 1995).It should be emphasized that Eq. (1) does not contain Newton’s gravitational constant G; in fact, this is thereason for the large numerical value in (1) with regard to relativistic standards. Another remarkable char-acteristic of relation (1) is the independence of the clock effect from the distance of the orbiting satellites, afeature that is to some extent reminiscent of the Aharonov-Bohm effect. Finally, the positive sign in Eq. (1)indicates that the corotating satellite is slower than the counterrotating one, apparently in contradiction tothe Machian idea of the dragging of inertial frames. More about these interesting peculiarities can be foundin Mashhoon et al., 1999.GRAVITATIONAL PERTURBATIONSIn principle, the validity of Eq. (1) can be tested by means of two clocks carried aboard two satellitesalong identical but opposite trajectories about the Earth, where each clock is read out when it has exactlycovered an azimuthal angle of 2pi. For an orbit of 7000 km radius, the satellite travels ∼ 0.04 mas in 200 ns,whence the accuracy in the determination of the azimuthal closure must be at least of the same order, whichmakes extremely high demands on the tracking facilities. However, assuming no systematic errors in theposition measurements, these requirements become less stringent with the increasing number of revolutionsbecause of the accumulation of the gravitomagnetic clock effect. Another source of errors is due to thegravitational and non-gravitational perturbations on the satellites (Lichtenegger et al., 2000). While thelatter may be avoided by means of drag-free satellites, the former must be carefully modeled in order toextract the gravitomagnetic effect out of the clock data. In the following, we will concentrate only on thevariation of the orbital period due to the static part of the Earth’s gravitational field (an analysis of thedynamical part will be given elsewhere).Eq. (1) applies to the difference in the sidereal orbital period of two oppositely circling satellites. Theangular velocity of a satellite with respect to a fixed system is the sum ω˙ + Ω˙ cos i + M˙ (see Iorio, 2000),where ω˙ and Ω˙ are the time rates of the argument of perigee and the line of nodes, respectively, M˙ is the(perturbed) mean motion and i denotes the orbital inclination. Upon expansion of the terrestrial gravityfield in terms of spherical harmonics, the perturbation equations for the angular orbital elements read (e.g.Seeber, 1989)dωnmpqdt=GMRnEn¯an+3(√1− e2eFnmpG′npq −cot i√1− e2F′nmpGnpq)Snmpq ,dΩnmpqdt=GMRnEn¯an+3F ′nmpGnpqSnmpq√1− e2 sin i , (2)dMnmpqdt= n¯+GMRnEn¯an+3FnmpSnmpq[2(n+ 1)Gnpq − 1− e2eG′npq].Here, n and m are degree and order of the expansion, while p and q are summation indices running from0 to n and −∞ to ∞, respectively. Further, a, e and n¯ denote the semi-major axis, eccentricity and theunperturbed mean motion of the satellites, and ME and RE are the mass and mean radius of the Earth.Finally, Snmpq(ω,Ω,M) is a periodic function of the angular coordinates and Fnmp(i) and Gnpq(e) representthe inclination and eccentricity functions, respectively, where a prime indicates differentiation with respectto the argument (Kaula, 1966). Secular perturbations are induced by the even zonal harmonics and affectthe variables ω, Ω and M while leaving a, e and i unchanged, i.e. keeping the right hand side of (2)constant. Because Gnpq(e) represents an infinite series in e with its lowest power equal to |q|, for circularorbits Gnpq(e = 0) = δ0q, where δij is the Kronecker delta. Thus, to first order, the orbital period modifieddue to secular perturbations becomes in the limit e→ 0 by means of Eqs. (2)Pn0p0 ≃ 2pin¯[1− 2(n+ 1)RnEanFn0pCn0].In the following, in accordance with Eq. (1), we will consider orbits with e = i = 0 only. The most relevantperturbation by far is due to C20, i.e. the lowest zonal harmonic, resulting in a difference with respect tothe unperturbed period of ∆P sec20 ≃ −15.7 s for an orbit of 7000 km radius. Because the time dependence ofEqs. (2) is contained in the function S, we can estimate an upper limit for the change in the revolution timedue to periodic perturbations by replacing S with its amplitude (C2nm + S2nm)1/2 and again calculating thedifference from the unperturbed motion. The first non-vanishing periodic contribution is due to n = m = 2,having an ”amplitude” of |∆Pmax22 | ≃ 0.158 s with a period of the perturbation of PPert22 ≃ 52 min for aprograde orbit. In fact, only those combinations of n and m do not vanish for which the sum n+m is even.Based on NASA’s EGM96 gravitational field model (Lemoine et al., 1998), Table 1 lists all non-vanishingsecular contributions to the orbital period as well as the amplitudes (in conformity with the above meaning)and periods of the sectorial and tesseral perturbations up to order and degree 6.n,m 2,0 2,2 3,1 3,3 4,0 4,2 4,4∆P secnm(s) -15.7 -0.024|∆Pmaxnm |(s) 0.158 0.117 0.117 0.051 0.032PPert+nm (s) 3117 6233 2078 3117 1558n,m 5,1 5,3 5,5 6,0 6,2 6,4 6,6∆P secnm(s) -0.008|∆Pmaxnm |(s) 0.008 0.038 0.071 0.029 0.040 0.027PPert+nm (s) 6233 2078 1247 3117 1558 1039Table 1. Secular and periodic changes in the orbital period due to the gravitational field up to degree andorder 6 for an orbital radius of 7000 km (the index + denotes a prograde orbit).According to Table 1, the perturbation in the orbital period of a satellite due to the nonsphericity of theEarth can be up to 8 orders of magnitude larger than the gravitomagnetic ”perturbation”. The accuratemodeling of the gravitational perturbations of the Earth is limited by the uncertainty in the harmoniccoefficients Cnm and Snm. Figure 1a illustrates the mismodeling δP of the period induced by the secularperturbations up to degree 100, based on the accuracy of the even zonal harmonic coefficients given in theEGM 96 model. The three lines correspond to an orbital radius of 7000, 8000 and 12000 km, respectively.As can be seen, for a near Earth orbit the accuracy of about the first hundred spherical harmonics is lessthan required, while for a 12000 km orbit only the accuracy of the coefficients up to degree ∼ 20 are ofimportance. Figure 1b shows the maximum mismodeling of the periodic perturbations due to the uncertaintyFigure 1: Mismodeling of the period for three different orbital heights: (a) error due to the even zonalperturbations Cn0, (b) error due to the uncertainty in the coefficients Cn2 and Sn2.of the harmonic coefficients Cn2 and Sn2. For a 7000 km orbit, the error is up to 1000 times larger than theeffect being studied and decreases to a factor of 100 at a distance of 12000 km. However, in spite of theselarge errors, the influence of the periodic perturbations is of less significance because they will cancel outas well as will be masked by the clock effect after a sufficient number of revolutions due to its accumulativecharacter.SUMMARYThe gravitomagnetic clock effect, which involves a coupling between the orbital motion of a satellite andthe rotation of the Earth, can be considered as an interesting alternative way to verify the existence of thegravitomagnetic field predicted by General Relativity. We have examined the influence of the static part ofthe gravitational field of the Earth on the period of a satellite and have estimated, based on the EGM96gravity field model, the expected mismodeling with respect to the observation of the gravitomagnetic clockeffect. While periodic perturbations, which involve a mismodeling of 100-1000 times larger than required,may cancel out, secular perturbations must be handled carefully. Assuming an experimental error of ∼ 10%,we find the residual uncertainty exceeding the required accuracy by a factor of 100. Thus, a successful obser-vation of the clock effect has to await a distinct improvement in the determination of the gravitational field ofthe Earth. This may be expected within a couple of years based on the data of the upcoming GOCE mission.ACKNOWLEDGMENTWe thank Lorenzo Iorio for helpful comments.REFERENCESCiufolini, I., E. Pavlis, F. Chieppa, E. Fernandes-Vieira, and J. Pe´rez-Mercader, Test of General Relativityand Measurement 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).Gronwald, F., E. Gruber, H. Lichtenegger, and R.A. Puntigam, Gravity Probe Clock - Probing the Gravito-magnetic Field of the Earth by Means of a Clock Experiment, in Fundamental Physics in Space, pp.29–37, ESA SP-420 (1997).Iorio, L., Satellite gravitational orbital perturbations and the gravitomagnetic clock effect, Int. J.Mod. Phys.D, in press (gr-qc/0007014), (2000).Kaula, W.M., Theory of Satellite Geodesy, 124 pp., Blaisdell Publishing Company, Waltham, (1966).Lemoine, F.G., et al., The Development of the Joint NASA GSFC and the National Imagery MappingAgency (NIMA) Geopotential Model EGM96, NASA/TP-1998-206861 (1998).Lichtenegger, H.I.M., F. Gronwald and B. Mashhoon, On Detecting the Gravitomagnetic Field of the Earthby Means of Orbiting Clocks, Adv. Space Res., 25, 1255–1258 (2000).Mashhoon, B., On the gravitational analogue of Larmor’s theorem, Phys. Lett. A, 173, 347–354 (1993).Mashhoon, B., F. Gronwald and D.S. Theiss, On measuring gravitomagnetism via spaceborne clocks: agravitomagnetic clock effect, Ann.Physik, 8, 135–152 (1999).Mashhoon, B., F. Gronwald and H. Lichtenegger, Gravitomagnetism and the Clock Effect, in TestingRelativistic Gravity in Space, edited by C. La¨mmerzahl, C.W. F. Everitt and F.W. Hehl, Springer,Berlin (2001).Seeber, G., Satellitengeoda¨sie, 489 pp., Walter de Gruyter, Berlin (1989).Will, C.M., Testing Machian Effects in Laboratory and Space Experiments, in Mach’s Principle: FromNewton’s Bucket to Quantum Gravity, volume 6 of Einstein Studies, edited by J. Barbour and H. Pfister,pp. 365–385, Birkha¨user, Boston (1995).

Some aspects on the observation of the gravitomagnetic clock effect

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