Gravimetry, Relativity, and the Global Navigation Satellite Systems

— Lesson: Flash Introduction to Gravimetry —

Albert Tarantola

March 14, 2005

Abstract With the identiﬁcation of the (geometrical) Einstein tensor of space-time to the (material) stress- energy tensor, the contracted Bianchi identities lead, when using three-dimensional notations, to two ‘conservation equations’, for the mass and for the linear momentum. Newton’s gravitational potential V then appears as the logarithm of the square root of g00 , the time-time component of the space-time metric. The goal of traditional Earth’s gravimetry is to evaluate the coefﬁcients of the development of V in spherical harmonics, with ad-hoc methods to account for the time variations. Gravimetry satellites become more and more sophisticated: after champ (with an ac- celerometer), and the grace duo, the goce satellite will soon be launched (three-axis gradiometry). From a more fundamental point of view, lisa may detect gravitational waves.

1 Contents

1 From Einstein to Newton (Quick and Dirty) 3

2 Gravity Field of the Earth 4 2.1 Common Terminology and Notations ...... 4 2.2 Spatial Variations ...... 5 2.3 Temporal Variations ...... 5

3 Champ, Grace, Goce. . . 11

4 Lisa 12

5 Clock Accuracy 13

6 Bibliography 13

A Field of a Spherical Mass 14

B Kerr Metric in Boyer-Lindquist Coordinates 14

C Grace Gravity Model (Geoid) 15

D Grace Gravity Model (Notes) 16

E Grace Gravity Model (First Coefﬁcients) 22

F Lense-Thirring 24

G Superconducting Gravimeters 26

2 1 From Einstein to Newton (Quick and Dirty)

In general relativity, the gravity field is the space time metric gαβ . This contrasts with Newton’s theory, where the gravity field is a scalar, the gravitation potential V . Let us made a quick and dirty development to see how we can see V from within general relativity. In an arbitrary metric manifold, if the metric gαβ is given, one can evaluate the Einstein tensor Eαβ , a purely geometrical tensor that satisfies the constraint (called the ‘contracted Bianchi identity’)

β ∇β Eα = 0 . (1)

General relativity arises when postulating that the Einstein tensor of the space-time manifold is proportional to the space-time stress-energy tensor tαβ :

8 π G β tαβ = Eα . (2) c2 Assume that, within the framework of general relativity, we are able to construct a space-time coordinate system that, such that, in a ﬁnite region of the space-time allows to write the metric as

2 g00 g0j ϕ 0 {gαβ} = = , (3) gi0 gi j 0 −γi j where γi j is a three-dimensional space metric, and write the stress-energy tensor as t00 t0j ρ p j {tαβ} = = , (4) ti0 ti j pi σi j where ρ is the mass-energy density, pi the linear momentum density, and σi j the three-dimensional stress tensor. The Bianchi identities then lead to the following equations

∂ρ 2 + ∇ pi = g pi ∂t i c2 i ∂ (5) ∂p 1 1 gi j i + ∇ σ j = ρ g + g σ j + p j , ∂t j i i c2 j i 2 ∂t where ∂V g = − ; V = logϕ . (6) i ∂xi Disregarding the terms in 1/c2 this gives ∂ ρ i ∂ + ∇i p = 0 t (7) ∂p i + ∇ σ j = ρ g . ∂t j i i The ﬁrst equation expresses the classical conservation of mass. The second equation expresses the classical conservation of linear momentum, and shows that there is the (three-dimensional) force density fi = ρ gi , (8)

3 well known from Newton’s gravitation theory. Newton’s gravity field gi is defined in terms of the gravitation potential as gi = −∂iV , and this is exactly the expression at left in equation 6. Replacing ϕ by its definition leads to √ V = log g00 , (9) i.e., Newton’s gravitation potential is the logarithm of the square root of g00 . Even in the situations where Newton’s gravitation theory applies, it is good to keep this relation in mind, just in case one discovers that clocks have an “odd behaviour” (from Newton’w point of view!).

2 Gravity Field of the Earth

Let’s forget general relativity, to see how Newton’s theory is used to describe the gravity ﬁeld of the Earth.

2.1 Common Terminology and Notations One calls force of gravity the resultant of the gravitational force and the centrifugal force. While an ob- server at the Earth’s surface senses the force of gravity, satellites only sense the gravitational force. The gravitational potential of a spherical distribution of mass is

G m V(r) = , (10) r where the gravitational constant has the value

G = 6.672 10−11 m3 kg−1 s−2 . (11)

The gravitational potential of an arbitrary distribution of mass is computed as

Z ρ(x0) V(x) = G dV(x0) . (12) k x − x0 k

The gravitational potential satisﬁes the Poisson equation

∆V = −4 π G ρ . (13)

In particular, outside a distribution os mass, where ρ = 0 , one has the Laplace equation

∆V = 0 . (14)

The gravitational acceleration is b = grad V . (15) The angular velocity of the Earth is, at present,

ω = 7.292 115 rad s−1 . (16)

Let p denote the distance to the rotation axis. The centrifugal acceleration z has the norm

z = ω2 p , (17)

4 and is directed outwards. One introduces the centrifugal potential

ω2 p2 Φ = Φ(p) = , (18) 2 and one has ∆Φ = 2 ω2 . One calls gravity acceleration the resultant of the gravitational acceleration b and the centrifugal acceleration z : g = b + z . (19) On the Earth’s surface, g is the direction of the “plumb line”. The gravity potential is, by deﬁnition, the sum

W = V + Φ . (20) then, the gravity acceleration is g = grad W , (21) and one obtains the generalized Poisson equation

∆W = −4 π G ρ + 2 ω2 . (22)

2.2 Spatial Variations The gravitational potential is written, in geographical coordinates {r, ϑ, λ} as

GM n V(r, ϑ, λ) = 1 + ∑ ∑ ( Cnm cos mλ + Snm sin mλ ) Pnm(cos ϑ) , (23) r n∞=1 m=0 where the Pnm(t) are the associated Legendre functions of the ﬁrst kind. The determination of the coefﬁ- cients Cnm and Snm determines the gravitational potential of the Earth. See appendices C–E for the grace model.

2.3 Temporal Variations For the description of the time variability, please have a look at the scans in ﬁgures 1–6, from Torge (1980). (These ﬁgures will not display correctly on a computer screen: for easy reading, they should be printed.)

Figure 1: Temporal variations of the gravity ﬁeld, from Torge (1980), with permission.

5 Figure 2: Temporal variations of the gravity ﬁeld, from Torge (1980), with permission.

6 Figure 3: Temporal variations of the gravity ﬁeld, from Torge (1980), with permission.

7 Figure 4: Temporal variations of the gravity ﬁeld, from Torge (1980), with permission.

8 Figure 5: Temporal variations of the gravity ﬁeld, from Torge (1980), with permission.

9 Figure 6: Temporal variations of the gravity ﬁeld, from Torge (1980), with permission.

10 3 Champ, Grace, Goce. . .

There have been many satellite mission to study the gravitational potential of the Earth. In the beginning the satellites were passive, like lageos (see appendix F). They are now becoming increasingly sophisticated. Figure 7 is from Felix Perosanz (CNES). Let us use it to rapidly describe three gravity missons (two recent, one yet to come).

La géodésie spatiale : Détermination de modèles de champ de gravité statiques et variables

•Orbite précise (GPS+accélérométre) •Analyse spectrale perturbation orbite ou méthode de intégrale de énergie •Gain dʼun ordre de grandeur sur la précision des modèles précédents

•Orbite précise (GPS+accélérométre) •Mesure distance relative bande K (qq microns !) •Analyse spectrale perturbation relative dʼorbite •Accès aux variations mensuelles du champ de gravité

•Orbite précise (GPS+traînée compensée) •Gradiomètre 3 axes •Cartographie a haute résolution du champ de gravité

Projets futurs: GRACE FO, MICROMEGA, LICODY…

Orbitographie précise de satellites en orbite basse Félix Perosanz Journée thématique « Galileo et la science », 11/06/04, CNES, Paris

Figure 7: A slide from Felix-Perosanz (CNES).

The champ satellite was precisely followed using the GPS system, and contained one accelerom- eter. The grace system consists in a pair of satellites following each other, Each satellite is similar to a champ satellite, but, with addition, their mutual distance is being measured with an accuracy of some microns. The next mission, goce, will embark a three-axis gradiometer (to measure ‘tidal forces’). We will come back to gradiometry at the end of this course.

11 4 Lisa

The following description comes from NASA web site.

The Laser Interferometery Space Antenna (LISA) is the first dedicated space-based gravitational wave observatory. So far the only space searches for gravitational waves have been made using measurements of radio signals, not optimized for gravitational wave searches, from spacecraft on their way to other planets. LISA will use an advanced system of laser interferometry and the most delicate measuring instruments ever made to directly detect gravitational waves. To accomplish its science objectives, the LISA mission depends on three core technologies: gravitational reference sensors, micronewton thrusters, and laser interferometry. These technologies will help to conteract the mission’s biggest challengedisturbances that mimic gravitational waves. Because LISA detects gravitational waves by measuring the change in distance between freely floating test masses, sources of both external and internal disturbance need to be eliminated or damped down to extremely low levelsmuch lower than what is needed by other missions. By minimizing such disturbances, motions that would imitate or mask the effect of gravitational waves are less likely to occur. Examples of external disturbances are the pressure from the light of the Sun, and its very small variations, the variable solar magnetic field, and Earth’s gravitational effects. To counteract the solar disturbances, the spacecraft structure will act as a shield to protect the test masses. And, LISA’s orbit, 20 degrees behind the Earth’s orbit of the Sun, will minimize the effects of the Earth’s gravity. Examples of internal disturbances are the interaction of the electrical field generated by the spacecraft computer acting on the test masses, effects from residual gas pressure near the test masses, and thermal radiation by the electrodes used to measure the spacecraft position. In order to minimize the effect of internal disturbances, the spacecraft must be controlled to follow the test masses with an accuracy of 10 nanometers, or about 1/100th of the wavelength of light! This spacecraft position-control operation, called ”drag-free,” is similar to the operation of low-Earth orbiting satellites that need to correct for the force due to the friction, or drag, of the Earth’s atmosphere. Disturbances are conteracted by micronewton thrusters. To keep external and internal forces on the spacecraft’s mass from affecting the test masses too much, the distance between the test masses is constantly moni- tored by LISA’s gravitational reference sensor. If this distance shifts, microthrusters fire to move the spacecraft back into position, away from the undisturbed test masses. Additionally, other specific design measures will counteract the possible disturbances. With the elimination of external and internal disturbances, LISA will be able to detect passing gravitational waves in low-frequency bands not previously possible. Another challenge is maintaining equal distances between LISA’s spacecraft as they fly in their triangular formation. Laser interferometry measurements are more difficult to make if the distances between pairs of spacecraft are not nearly equal. This formation was chosen because LISA needs to measure the distance between proof masses, separated by five million kilometers, with an accuracy of 10 picometers (10 millionths of a micron or half a billionth of an inch)! LISA’s advanced technologies will address this challenge and ensure the mission’s success.

Figure 8: The lisa triangle.

12 5 Clock Accuracy

It is the ever-increasing accuracy of clocks (see ﬁgure 9) that will force us to take relativity more seriously, and to pass from the classical ‘gravimetric’ vision of the neighborhood of the Earth to the fully relativistic point of view that this seminar shall propose.

Figure 9: The ever- increasing accuracy of clocks.

6 Bibliography

For classical gravimetry, see: Torge, W., 1980, Geodesy, Walter de Gruyter.

13 A Field of a Spherical Mass

Just to have the equations at hand, this is the exterior ﬁeld (space-time metric) of a spherical distribution of mass. First, the ﬂat space-time metric:

1 ds2 = dt2 − (dx2 + dy2 + dz2) (24) c2 1 ds2 = dt2 − dr2 + r2 (dθ2 + sin2 θ dϕ2) . (25) c2 And, now, the Schwarzschild metric:

r 1 dr2 2 = − S 2 − + 2 ( θ2 + 2 θ ϕ2) ds 1 dt 2 r d sin d (26) r c 1 − rS/r r g = 1 − S (27) tt r c2 c2 r 1 g ≈ ∇g ; g = S (28) Newton 2 tt Newton 2 r2

2 GM GM r = ; g = (29) S c2 Newton r2

B Kerr Metric in Boyer-Lindquist Coordinates

The Kerr metric describes space-time outside a rotating mass. Using Boyer-Lindquist coordinates it is written

2 M r 4 J r sin2θ ds2 = 1 − dt2 + dt dφ r2 + J2 cos2θ / M2 r2 + J2 cos2θ / M2

r2 + J2 cos2θ / M2 J2 cos2θ − dr2 − r2 + dθ2 (30) r2 − 2 M r + J2/M2 M2

J2 2 r J2 sin2θ/M − r2 + + sin2θ dφ2 , M2 r2 + J2 cos2θ / M2 where M is a mass and J is an angular momentum. When J = 0 , one recovers the Schwarzschild metric 2 M 1 ds2 = 1 − dt2 − dr2 − r2 (dθ2 + sin2θ dφ2) . (31) r 1 − 2 M/r

14 C Grace Gravity Model (Geoid)

See ﬁgure 10 for a representation of the geoid obtained using grace data.

Figure 10: The grace geoid, from the Potsdam group (see this ﬁgure in color!).

15 D Grace Gravity Model (Notes)

Figures 11–16 are scans of the document giving the details of the grace gravity model.

Figure 11: Notes for the grace gravity model.

16 Figure 12: Notes for the grace gravity model.

17 Figure 13: Notes for the grace gravity model.

18 Figure 14: Notes for the grace gravity model.

19 Figure 15: Notes for the grace gravity model.

20 Figure 16: Notes for the grace gravity model.

21 E Grace Gravity Model (First Coefﬁcients)

Here is the beginning of the list of the coefﬁcients themselves.

GRACE Gravity Model 02 (GGM02) - Released October 29, 2004

2 0 -4.8416938905481E-04 0.0000000000000E+00 6.1000E-11 0.0000E+00 -1. 2 1 -2.0458338184745E-10 1.3968195379551E-09 2.8000E-11 2.9000E-11 -1. 2 2 2.4393233001191E-06 -1.4002662003867E-06 3.1000E-11 3.1000E-11 -1. 3 0 9.5718508415439E-07 0.0000000000000E+00 1.1000E-11 0.0000E+00 -1. 3 1 2.0304752656064E-06 2.4817416903031E-07 1.6000E-11 1.6000E-11 -1. 3 2 9.0480066975068E-07 -6.1900441427103E-07 2.2000E-11 2.2000E-11 -1. 3 3 7.2128924247650E-07 1.4143556434052E-06 2.6000E-11 2.6000E-11 -1. 4 0 5.3999143526074E-07 0.0000000000000E+00 8.2000E-12 0.0000E+00 -1. 4 1 -5.3617583789434E-07 -4.7356802287476E-07 8.2000E-12 8.2000E-12 -1. 4 2 3.5051159931087E-07 6.6243944849186E-07 1.1000E-11 1.1000E-11 -1. 4 3 9.9085503541734E-07 -2.0097529442342E-07 1.6000E-11 1.6000E-11 -1. 4 4 -1.8846750474516E-07 3.0882228278756E-07 1.5000E-11 1.5000E-11 -1. 5 0 6.8715981001179E-08 0.0000000000000E+00 5.9000E-12 0.0000E+00 -1. 5 1 -6.2904051380481E-08 -9.4373263356928E-08 6.3000E-12 6.3000E-12 -1. 5 2 6.5210393080303E-07 -3.2334838450788E-07 7.6000E-12 7.6000E-12 -1. 5 3 -4.5187965449909E-07 -2.1500140801223E-07 1.2000E-11 1.2000E-11 -1. 5 4 -2.9533633996919E-07 4.9817834613976E-08 1.9000E-11 1.9000E-11 -1. 5 5 1.7479367861529E-07 -6.6937444851133E-07 1.8000E-11 1.8000E-11 -1. 6 0 -1.4994011973125E-07 0.0000000000000E+00 4.5000E-12 0.0000E+00 -1. 6 1 -7.5903534713091E-08 2.6516969820404E-08 4.6000E-12 4.6000E-12 -1. 6 2 4.8671477410889E-08 -3.7379063632786E-07 5.5000E-12 5.5000E-12 -1. 6 3 5.7235251140830E-08 8.9355908317172E-09 6.3000E-12 6.3000E-12 -1. 6 4 -8.6024306020762E-08 -4.7142492778928E-07 9.6000E-12 9.6000E-12 -1. 6 5 -2.6717044535375E-07 -5.3648870593762E-07 1.6000E-11 1.6000E-11 -1. 6 6 9.4667858117668E-09 -2.3740563878695E-07 1.2000E-11 1.2000E-11 -1. 7 0 9.0504630229132E-08 0.0000000000000E+00 3.3000E-12 0.0000E+00 -1. 7 1 2.8088898145081E-07 9.5119644062781E-08 3.4000E-12 3.4000E-12 -1. 7 2 3.3041560746535E-07 9.2985239787248E-08 3.9000E-12 3.9000E-12 -1. 7 3 2.5045166263142E-07 -2.1714748211003E-07 5.6000E-12 5.5000E-12 -1. 7 4 -2.7498938059553E-07 -1.2406704411494E-07 6.1000E-12 6.1000E-12 -1. 7 5 1.6601317328580E-09 1.7933888039325E-08 8.7000E-12 8.7000E-12 -1. 7 6 -3.5880775879961E-07 1.5179454735699E-07 1.4000E-11 1.4000E-11 -1. 7 7 1.5062884219099E-09 2.4116259277940E-08 1.4000E-11 1.4000E-11 -1. 8 0 4.9481334104780E-08 0.0000000000000E+00 2.8000E-12 0.0000E+00 -1. 8 1 2.3159979727734E-08 5.8896665124862E-08 2.9000E-12 2.9000E-12 -1. 8 2 8.0015152191272E-08 6.5278835020898E-08 3.2000E-12 3.2000E-12 -1. 8 3 -1.9378833774969E-08 -8.5977473175173E-08 3.6000E-12 3.6000E-12 -1. 8 4 -2.4436744168468E-07 6.9808169573661E-08 4.9000E-12 4.9000E-12 -1. 8 5 -2.5695381020603E-08 8.9195503483206E-08 5.7000E-12 5.7000E-12 -1. 8 6 -6.5962471839471E-08 3.0894479673499E-07 7.9000E-12 7.9000E-12 -1. 8 7 6.7261439849585E-08 7.4875670873080E-08 1.4000E-11 1.4000E-11 -1. 8 8 -1.2403873503204E-07 1.2055330686769E-07 9.5000E-12 9.6000E-12 -1. 9 0 2.8023188203329E-08 0.0000000000000E+00 2.4000E-12 0.0000E+00 -1. 9 1 1.4215021812885E-07 2.1399410624745E-08 2.5000E-12 2.5000E-12 -1. 9 2 2.1413308310842E-08 -3.1693757099588E-08 2.7000E-12 2.7000E-12 -1. 9 3 -1.6061777555625E-07 -7.4288065727748E-08 3.5000E-12 3.5000E-12 -1. 9 4 -9.3744494781812E-09 1.9903426483849E-08 3.7000E-12 3.7000E-12 -1.

22 9 5 -1.6310476153586E-08 -5.4041923229450E-08 5.1000E-12 5.1000E-12 -1. 9 6 6.2782214351230E-08 2.2295178138467E-07 6.0000E-12 6.0000E-12 -1. 9 7 -1.1797705231858E-07 -9.6928434857123E-08 7.9000E-12 7.9000E-12 -1. 9 8 1.8814003080096E-07 -3.0006845470535E-09 1.2000E-11 1.2000E-11 -1. 9 9 -4.7556714055660E-08 9.6880058387687E-08 9.6000E-12 9.6000E-12 -1. 10 0 5.3320010526550E-08 0.0000000000000E+00 2.0000E-12 0.0000E+00 -1. 10 1 8.3764843251410E-08 -1.3109151284691E-07 2.1000E-12 2.1000E-12 -1. 10 2 -9.3986202660537E-08 -5.1279822748794E-08 2.2000E-12 2.2000E-12 -1. 10 3 -7.0166481788361E-09 -1.5414522056325E-07 2.5000E-12 2.5000E-12 -1. 10 4 -8.4472635586329E-08 -7.9031973795022E-08 3.0000E-12 3.0000E-12 -1. 10 5 -4.9292034854895E-08 -5.0611717909905E-08 3.3000E-12 3.3000E-12 -1. 10 6 -3.7585195034396E-08 -7.9769574266997E-08 4.6000E-12 4.6000E-12 -1. 10 7 8.2601322177926E-09 -3.0472813339748E-09 5.5000E-12 5.5000E-12 -1. 10 8 4.0596116481898E-08 -9.1718696376024E-08 7.1000E-12 7.1000E-12 -1. 10 9 1.2538272379586E-07 -3.7954367789185E-08 1.2000E-11 1.2000E-11 -1. 10 10 1.0042456986404E-07 -2.3859595666514E-08 8.1000E-12 8.1000E-12 -1. 11 0 -5.0772496297652E-08 0.0000000000000E+00 1.9000E-12 0.0000E+00 -1. 11 1 1.5603252740856E-08 -2.7119062596280E-08 1.9000E-12 1.9000E-12 -1. 11 2 2.0113610023582E-08 -9.9008467215296E-08 2.1000E-12 2.1000E-12 -1. 11 3 -3.0581202091025E-08 -1.4884347611180E-07 2.5000E-12 2.5000E-12 -1. 11 4 -3.7952641039471E-08 -6.3769759692111E-08 2.7000E-12 2.7000E-12 -1. 11 5 3.7421284143008E-08 4.9590085390006E-08 3.4000E-12 3.4000E-12 -1. 11 6 -1.5589638020177E-09 3.4274419785441E-08 3.8000E-12 3.8000E-12 -1. 11 7 4.6579602935627E-09 -8.9819491456863E-08 5.1000E-12 5.1000E-12 -1. 11 8 -6.3030987247306E-09 2.4544222515068E-08 5.4000E-12 5.4000E-12 -1. 11 9 -3.1069260209285E-08 4.2065869634528E-08 7.4000E-12 7.4000E-12 -1. 11 10 -5.2252693035196E-08 -1.8423731781649E-08 9.9000E-12 9.9000E-12 -1. 11 11 4.6240805736467E-08 -6.9668542043034E-08 9.2000E-12 9.2000E-12 -1. 12 0 3.6436967894238E-08 0.0000000000000E+00 1.8000E-12 0.0000E+00 -1. 12 1 -5.3587669208192E-08 -4.3163587100214E-08 1.8000E-12 1.8000E-12 -1. 12 2 1.4267337095997E-08 3.1093083556009E-08 1.9000E-12 1.9000E-12 -1. 12 3 3.9615951283603E-08 2.5063409200897E-08 2.1000E-12 2.1000E-12 -1. 12 4 -6.7736670283692E-08 3.8405890605906E-09 2.4000E-12 2.4000E-12 -1. 12 5 3.0879106734968E-08 7.5901279330467E-09 2.7000E-12 2.7000E-12 -1. 12 6 3.1334008647374E-09 3.8974784317029E-08 3.3000E-12 3.3000E-12 -1. 12 7 -1.9044941634320E-08 3.5732805442333E-08 3.6000E-12 3.6000E-12 -1. 12 8 -2.5889757115756E-08 1.6936326994120E-08 5.2000E-12 5.2000E-12 -1. 12 9 4.1919489963425E-08 2.4964248875423E-08 5.9000E-12 5.9000E-12 -1. 12 10 -6.2010070538131E-09 3.0946924871442E-08 7.3000E-12 7.4000E-12 -1. 12 11 1.1361953743426E-08 -6.3910578294950E-09 1.1000E-11 1.1000E-11 -1. 12 12 -2.4302921673268E-09 -1.1104248894541E-08 8.1000E-12 8.1000E-12 -1. ...

23 F Lense-Thirring

Ciufolini and Pavlis claim to have measured the lense-Thirring effect using data from lageos. See ﬁgures 17–19.

Figure 17: Lense-Thirring.

24 Figure 18: Lense-Thirring.

Figure 19: Lense-Thirring.

25 G Superconducting Gravimeters

Imanishi et al. (2004) show how superconducting gravimeters may reveal submicrogal seismic gravity changes (see ﬁgures 20 and 21).

Figure 20: Imanishi et al.

Figure 21: Imanishi et al.