Roberto Peron IFSI-INAF Email: [email protected].

GRAVITOMAGNETISM AND OTHER PHYSICS WITH THE LAGEOS SATELLITES

Roberto PeronIFSI-INAFEmail: [email protected]

INTRODUCTION (MOTIVATIONS)

INTRODUCTION (MOTIVATIONS)

Near-Earth space (spacetime) is a good place to perform tests on theories about gravitation

Earth Schwarzschild radius 1 cm

The effects searched for are by now relevant for current technology: think about GPS!

2

2

c

GM

INTRODUCTION (MOTIVATIONS)

What do we need in order to perform good science?

A theory: Schwarzschild, Kerr (gravitomagnetism) exact solutions sufficiently general to be descriptive and predictive

Contat points with experiment: weak field and slow motion, PPN formalism

A probe: test masses

LAGEOS SATELLITES AND SLR

LAGEOS AND SLR

General relativity (geometrodynamics) implies a continuous feedback between geometry and mass-energy

(nonlinearity)

Practical needs often force to “hold on something”

TEST MASS• No electric charge• Gravitational bounding energy

negligible with respect to rest mass-energy

• Angular momentum negligible• Sufficiently small to neglect tidal

effects

THE MOONThe smallness of a test mass depends on the scale under consideration

LAGEOS and SLR

CASSINIA test mass in the outer solar system

LAGEOS and SLR

BEPICOLOMBOA future test mass pretty close to the Sun

LAGEOS and SLR

LAGEOS IIThe LAGEOS satellite are probably the closest to the ideal concept of a test mass

LAGEOS and SLR

LAGEOS AND SLR

LAGEOS LAGEOS II

– COSPAR ID 7603901 9207002

– Launch date 4 May 1976 22 October 1992

– Diameter 60 cm 60 cm

M Mass 406.965 kg 405.38 kg

– Retroreflectors 426 CCR 426 CCR

a Semimajor axis 1.2286 · 107 m 1.2155 · 107 m

e Eccentricity 0.0045 0.0135

I Inclination 109.84° 52.64°

– Perigee height 5.86 · 106 m 5.62 · 106 m

P Period 225 min 223 min

n Mean motion 4.654 · 10−4 s−1 4.696 · 10−4 s−1

Node rate 0.34266° d−1 -0.62576641° d−1

Perigee rate -0.21338° d−1 0.44470485° d−1

LAGEOS AND SLR

Satellite Laser Ranging (SLR)A laser pulse from a ground station is sent to the satellite, where it is reflected back in the same direction from optical elements called Cube Corner Retroreflectors (CCR)The precision of this technique is noteworthy ( 1 mm)

Matera MLRO

CCRilrs.gsfc.nasa.gov

INTERNATIONAL LASER RANGING SERVICE (ILRS) NETWORK

ilrs.gsfc.nasa.gov

LAGEOS and SLR

MAIN SPACE GEODETIC NETWORKSilrs.gsfc.nasa.gov

LAGEOS and SLR

GRAVITOMAGNETISM (IN WEAK-FIELD AND SLOW-MOTION)

GRAVITOMAGNETISM

Moving (rotating) masses: what do they do? - Spacetime

dtdr

Jddrdr

r

Mdt

r

Mds 222222

122 sin

4sin

21

21

Kerr metric in weak-field (it describes in an approximate way the spacetime around a rotating mass)

g

!!!Mach

?

GRAVITOMAGNETISM

Moving (rotating) masses: what do they do? - Spacetime

TRgR 82

1

hg

ii vh 1602

weak field

Lorentz gauge

hH

ih0 Gravitomagnetic potential

Gravitomagnetic field

Defined by analogy with electromagnetic case

GRAVITOMAGNETISM

Moving (rotating) masses: what do they do? - Geodesics

02

2

2

d

xd

d

dx

d

xd

Slow-motion

H

dt

xdGm

dt

xdm

2

2

Gravitoelectric field

Gravitomagnetic contribution

Thus mass-energy currents influence the motion of test

masses:Gravitomagnetism

GRAVITOMAGNETISM

32)(x

xJxh

3

ˆ)ˆ(32

x

xxJJH

3

ˆ)ˆ(3

x

xxJJ

Spherically symmetric rotating mass-energy distribution (J is the angular momentum associated to the distribution)

A gyroscope in a gravitomagnetic field precesses

Dragging of inertial frames

GRAVITOMAGNETISM

Obtain a solution

Celestial mechanics tools

• Osculating ellipse (Keplerian elements)• Perturbation first-order analysis (Lagrange and

Gauss equations)

• Periodic effects• Secular effects (

t)

GRAVITOMAGNETISM

Lagrange perturbation equations

a

R

nae

R

ena

en

dt

dM

e

R

ena

e

I

R

ena

I

dt

d

I

R

Ienadt

d

R

Iena

R

ena

I

dt

dI

R

ena

e

M

R

ena

e

dt

de

M

R

nadt

da

21

1

1

cot

sin1

1

sin1

1

1

cot

11

2

2

2

2

2/12

2/122

2/122

2/1222/122

2

2/12

2

2

GRAVITOMAGNETISM

Gauss perturbation equations

ttdtnM

dI

dt

deR

a

r

nadt

d

fIrH

W

e

ufTfR

nae

e

dt

d

frIH

W

dt

d

frH

W

dt

dI

ufTfRna

e

dt

de

feTfendt

da

0

2/12

2/12

2/12

2/12

2/12

dtcos1

2

sincot1

sinsincos

1

sinsin

cos

coscossin1

cos1sinRe1

2

GRAVITOMAGNETISM

2/3232 1

2

eac

GJTL

Ieac

GJTL cos1

62/3232

Secular effects on longitude of ascending node and argument of perigee

J. Lense and H. Thirring, 1918

LAGEOS

LAGEOS II

30.66 31.51

31.31 -57.35

Values in mas1 mas = 2.8 ∙ 10-5 °

TL

TL

PARAMETER ESTIMATION

W

P

CM

COf

dPz

j

iij

iii

ii

PARAMETER ESTIMATION

Differential correction procedure

j ij

j

iii dOdP

P

CCO

fWMMWMz TT 111

Corrections to the models parameters

Residuals

Observation equations

Least-squares (normal equations)

Partials

Covariance matrix

PROCEDURE SKETCH

Parameter estimation

MODELS

MODELS

The analysis of experimental data to obtain the properties of a physical system requires models

System dynamicsMeasurement procedure(Reference frame)

The availability of good experimental data implies taking out a lot of “noise” in order to reach the phenomenology of interest – many orders of magnitude, in case of relativistic effects

MODELS

• Geopotential (static part)• Solid Earth and ocean tides / Other temporal

variations of geopotential• Third body (Sun, Moon and planets)• de Sitter precession• Deviations from geodetic motion• Other relativistic effects• Direct solar radiation pressure• Earth albedo radiation pressure• Anisotropic emission of thermal radiation due to

visible solar radiation (Yarkovsky-Schach effect)• Anisotropic emission of thermal radiation due to

infrared Earth radiation (Yarkovsky-Rubincam effect)

• Asymmetric reflectivity• Neutral and charged particle drag

Gravitational

Non-gravitational

MODELS

Cause Formula Acceleration (m s-2)

Earth’s monopole 2.8

Earth’s oblateness 1.0 ∙ 10-3

Low-order geopotential harmonics

6.0 ∙ 10-6

High-order geopotential harmonics

6.9 ∙ 10-12

Perturbation due to the Moon

2.1 ∙ 10-6

Perturbation due to the Sun

9.6 ∙ 10-7

General relativistic correction

9.5 ∙ 10-10

Table taken from A. Milani, A. Nobili, and P. Farinella, Non–gravitational perturbations and satellite geodesy, Adam Hilger, 1987

2r

GM

20

2

23 J

r

R

r

GM

224

2

3 Jr

RGM

18,1820

18

19 Jr

RGM

rr

GM

Moon

Moon3

2

rr

GM

Sun

Sun3

2

rc

GM

r

GM 122

MODELS

Cause Formula Acceleration (m s-2)

Atmospheric drag 3 ∙ 10-12

Solar radiation pressure

3.2 ∙ 10-9

Earth’s albedo radiation pressure

3.4 ∙ 10-10

Thermal emission 1.9 ∙ 10-12

Dynamic solid tide 3.7 ∙ 10-8

Dynamic ocean tide 0.1 of the dynamic solid tide

3.7 ∙ 10-9

Reference system: non-rigid Earth nutation (fortnightly term)

0.002 arsec in 14 days

3.5 ∙ 10-12

Table taken from A. Milani, A. Nobili, and P. Farinella, Non–gravitational perturbations and satellite geodesy, Adam Hilger, 1987

2

2

1V

M

ACD

cM

A Sun

2

r

RA

cM

A Sun

09

4

T

T

cM

A Sun

4

32

23 r

R

r

R

r

GMk

MoonMoon

Moon

MODELS

The Earth is not a sphere!

Spherical harmonics expansion

1 0

sincoscos1)(l

l

mlmlmlm

l

mSmCPr

R

r

GMrU

Spherical harmonics classification

Zonal

Tesseral

Sectorial

0m

0m

lm

lm

0m

MODELS

Quadrupole perturbation (l = 2, m = 0) to first order

2/32

22

20

22

22

20

22

2

20

1

cos31

4

3

1

cos51

4

3

1

cos

2

3

0

0

0

e

I

a

RnCn

dt

dM

e

I

a

RnC

dt

d

e

I

a

RnC

dt

d

dt

dIdt

dedt

da

MODELS

Some geopotential models

Model Data type

Maximum degree

JGM-3 Combined 70

GRIM5-S1 Satellite 95

GRIM5-C1 Combined 120

OSU89A/B Combined 360

EGM96 Combined 360

EIGEN-2 Satellite 120

EIGEN-GRACE02S Satellite 150

GGM02S Satellite 160

GEOID (EIGEN-GRACE02S)The geoid is a gravitational equipotential surface, taken as reference surface (“sea level”); It differs in general from a rotation surface, like the reference ellipsoid

Models

GRAVITY ANOMALIES (EIGEN-GRACE02S)The gravity anomalies are the difference between the real gravity field and that of a reference body (rotation ellipsoid)

Models

MODELS

l

mlmlml SC

lC

0

222

12

1

4

102 10

7.0l

Cl

The degree variance is useful when comparing various geopotential solutions

Its behaviour is well described by the so-called Kaula’s rule

A similar rule seems to be valid also for the Moon and the other terrestrial planets

EGM96 AND KAULA’S RULEEarth geopotential degree variance is well approximated by Kaula’s rule

Models

SIGNAL-TO-NOISE RATIO FOR EGM96The signal-to-noise ratio indicates how well the signal is recognizable from the noise

Models

COMPARISON BETWEEN EARTH AND MOONThough similar in behaviour, Earth and Moon gravity potentials differ in the way the power is distributed at the various wavelengths

Models

COMPARISON BETWEEN VARIOUS GEOPOTENTIAL MODELS

The various geopotential solutions differ strongly in the uncertainty associated to the harmonic coefficients

Models

MODELS

Direct solar radiation pressure

It is due to reflection-diffusion-absorption of solar photons from the spacecraft surface

sr

AU

mc

ACa R ˆ

12

• The strongest among the non-gravitational perturbations

• Well modeled for LAGEOS (though the CR estimate could be biased due to some other not modeled signal)

TOTAL SOLAR IRRADIANCEThe energy flux from the Sun varies with a periodicity of about 11 years (Solar Cycle); plot from www.pmodwrc.ch/pmod.php?topic=tsi/composite/SolarConstant

Models

MODELS

Yarkovsky-Schach effect

It is due to infrared radiation anisotropically emitted from the satellite (warmed by the Sun)

SaYar ˆcos

TTRmcir 3

02

9

16

• Effective on argument of perigee behaviour• Difficult modelization (the acceleration depends

on S)

SPIN EVOLUTION OF LAGEOSFarinella–Vokhroulicky–Barlier model

Models

SPIN EVOLUTION OF LAGEOS IIFarinella–Vokhroulicky–Barlier model

Models

MODELS

“Inertial” reference

frame

ICRF

Precession

Nutation

Length of Day

Pole motion

ITRF

POLE MOTIONIERS data (EOP 05 C04)

Models

LENGTH OF DAY VARIATIONIERS data (EOP 05 C04)

Models

DATA ANALYSIS (EXTRACTING PHYSICS)

DATA ANALYSIS

• Tracking data: Normal Point from ILRS• Software: Geodyn II (NASA GSFC)• Arc length: 15 days• Estimate: state vector and other model

parameters• Further information: residuals

RESIDUAL = OBSERVED - CALCULATED

DATA ANALYSIS

Feature Model

Geopotential (static part) EGM96, EIGEN-2, GGM01S, EIGEN-GRACE02S(among others)

Geopotential (tides) Ray GOT99.2

Third body JPL DE-403

Relativistic corrections PPN

Direct solar radiation pressure

Cannonball

Albedo radiation pressure Knocke–Rubincam

Yarkovsky–Rubincam effect GEODYN model

Spin axis Farinella–Vokhroulicky–Barlier

Stations positions (ITRF) ITRF2000

Ocean load Scherneck with GOT99.2 tides

Pole motion IERS EOP

Earth rotation IERS EOP

DATA ANALYSIS

Element

Value Uncertainty

a 12265733.6258 m

0.1616 m

e 0.005137677709

0.000000008907

I 109.891430986°

0.000001159°

314.057527507°

0.000001163°

45.867206979° 0.000158387°

M 212.542258784°

0.000157151°

LAGEOS initial conditions estimate for 15-day arc with epoch 3 January 1993 (MJD 48990)

DATA ANALYSIS

Element

Value Uncertainty

a 12160019.9315 m

0.0066 m

e 0.013277589498

0.000000010922

I 52.622724892° 0.000001051° 68.835933790° 0.000001258° 244.234411803

°0.000061024°

M 51.909462952° 0.000063071°

LAGEOS II initial conditions estimate for 15-day arc with epoch 3 January 1993 (MJD 48990)

LAGEOS POST-FIT RMS (EGM96)

Data analysis

LAGEOS II POST-FIT RMS (EGM96)

Data analysis

LAGEOS POST-FIT RMS (VARIOUS MODELS)

Data analysis

LAGEOS II POST-FIT RMS (VARIOUS MODELS)

Data analysis

Data analysis – Residuals

Data analysis – Residuals

DATA ANALYSIS – LENSE-THIRRING

IIII

I

ITL

IIII

ITL

I N

N

N

N

2

2

2

2

The recent geopotential models make critical in the error budget only the uncertainty associated with C20 (Earth quadrupole)

Two-node combination to overcome this problem

SECULAR TREND IN THE COMBINED INTEGRATED NODAL RESIDUALS: EIGEN-GRACE02SCiufolini, Pavlis and Peron, New Astron. 11, 527 (2006)

Data analysis – Lense-Thirring

SECULAR TREND IN THE COMBINED INTEGRATED NODAL RESIDUALS: VARIOUS MODELSA peculiar error estimate is associated to each analysis

Data analysis – Lense-Thirring

DATA ANALYSIS – ESTIMATION BIASES

In common practice often empirical accelerations are included in the modelization setup to take into account small not modeled perturbations

Fit improving Estimation biases risk

Constant part ∼ 10−12 m/s2

Possible correlation between CR and empirical accelerations coefficients

PROBABLY FALSE SIGNAL

Data analysis – Estimation biases

TRUE SIGNAL?Lucchesi et al., Plan. Space Sci. 52, 699 (2004)

Data analysis – Estimation biases

NEW ESTIMATE OF YARKOVSKY–SCHACH EFFECT AMPLITUDE

Lucchesi et al., Plan. Space Sci. 52, 699 (2004)

Data analysis – Yarkovsky-Schach effect

ANOMALOUS NODE RESIDUALS BEHAVIOUR (LAGEOS)

The reported secular trend for the integrated residuals of the longitude of the LAGEOS ascending node abruptly changes from 0.026 mas d−1 to 0.24 mas d−1

Data analysis – Unexpected results

ANOMALOUS NODE RESIDUALS BEHAVIOUR (LAGEOS II)

The reported secular trend for the integrated residuals of the longitude of the LAGEOS II ascending node abruptly changes from -0.0079 mas d−1 to -0.53 mas d−1

Data analysis – Unexpected results

DATA ANALYSIS – UNEXPECTED RESULTS

Geopotential harmonics coefficients change in time:• Tides• Secular variations (e.g. postglacial rebound)• Mass transport (e.g. oceans ↔ atmosphere)

This variation seems to be due to an abrupt change in the quadrupole rate (Cox and Chao, Science 297, 831 (2002); Ciufolini, Pavlis and Peron, New Astron. 11, 527 (2006))

The causes are uncertain:• Mantle?• Tides?In any case, this implies a net mass transfer from polar to equatorial regions

READINGS (MINIMAL SET)

READINGS (MINIMAL SET)

D. W. Sciama, The Unity of the Universe, Faber & Faber, 1959C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman and Co., 1973I. Ciufolini and J. A. Wheeler, Gravitation and inertia, Princeton University Press, 1995B. Bertotti, P. Farinella, and D. Vokrouhlický, Physics of the Solar System — Dynamics and Evolution, Space Physics, and Spacetime Structure, Kluwer Academic Publishers, 2003A. Milani, A. Nobili, and P. Farinella, Non–gravitational perturbations and satellite geodesy, Adam Hilger, 1987O. Montenbruck and E. Gill, Satellite Orbits — Models, Methods and Applications, Springer, 2000C. M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Relativity 9, (2006), 3, http://www.livingreviews.org/lrr-2006-3D. McCarthy, and G. Petit (eds.), IERS Conventions (2003), IERS, 2004