l. iorio- recent attempts to measure the general relativistic lense-thirring effect with natural and...

8/3/2019 L. Iorio- Recent Attempts to Measure the General Relativistic Lense-Thirring Effect with Natural and Artificial Bodies in the Solar System

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The Lense-Thirring Effect

The Systematic Error of Gravitational Origin

Our Results

Solar and Martian Frame-Dragging and its Measurability

Summary

Recent Attempts to Measure the General

Relativistic Lense-Thirring Effect with Natural

and Artificial Bodies in the Solar System

L. Iorio

INFN-Sezione di Pisa

Fifth International School on Field Theory and Gravitation,

20-24 April 2009, Cuiab, Mato Grosso, MT, Brasil

L. Iorio Attempts to Measure Frame-Dragging in the Solar System


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Our Results


Summary

Outline


Gravitomagnetic Orbit Precessions

Attempts with the Spinning Earth, Sun and Mars

The Systematic Error of Gravitational OriginThe Effect of the Oblateness of the Central Body

The Corrupting Bias of the Terrestrial Even Zonals

How to Assess the Impact of the Even Zonals?

Our Results

A More Conservative Approach

Error Budgets Revisited


Summary



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A More Conservative ApproachError Budgets Revisited


Summary



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The Impact of Rotation of a Central Body on the Orbit

of a Test Particle in General Relativity

In GR a rotating body with spin S generates a gravitomagnetic

field [Iorio 2007] whose effects on the orbit of a test particle are

secular precessions of the node and the pericentre

LT =2GS

c2a3(1 e2)3/2, LT =

2GScos i

c2a3(1 e2)3/2

asemi-major axis of the orbit (fixes its size)

e eccentricity of the orbit (fixes its shape)

i inclination of the orbit to the equatorial plane of the

central body (i = 0 equatorial orbit; i = 90 polar orbit)



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Figure: Keplerian ellipse. The orbital elements , , i can be viewed as thethree (constant) Euler angles fixing the configuration of a rigid body, i.e. the

orbit which in the unperturbed Keplerian case does change neither its shape

nor its size, in the inertial {X, Y, Z} space.

S

i

x

z

M

f

w

W

P

m

y


Th L Thi i Eff


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Th L Thi i Eff t


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Measuring Terrestrial and Solar Frame-Dragging

First tests with the passive geodynamics satellites

LAGEOS and LAGEOS II tracked with the Satellite Laser

Ranging (SLR) technique in the gravitational field of the

spinning Earth. LARES should be launched by ASI at theend of 2009-beginning of 2010. Systematicuncertainties

due to mismodelled competing forcesare more important

than measurement errors

Tests with Venus [Iorio 2008a] and Mars Global Surveyor[Iorio 2006], in the gravitational fields of the spinning Sun

and Mars. Measurement errorsare, at present, perhaps

more important than systematicuncertainties due to

mismodelled competing forces


The Lense Thirring Effect


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Our Results


Summary



Measuring Terrestrial and Solar Frame-Dragging

First tests with the passive geodynamics satellites

LAGEOS and LAGEOS II tracked with the Satellite Laser

Ranging (SLR) technique in the gravitational field of the

spinning Earth. LARES should be launched by ASI at theend of 2009-beginning of 2010. Systematicuncertainties

due to mismodelled competing forcesare more important

than measurement errors

Tests with Venus [Iorio 2008a] and Mars Global Surveyor

[Iorio 2006], in the gravitational fields of the spinning Sun

and Mars. Measurement errorsare, at present, perhaps

more important than systematicuncertainties due to

mismodelled competing forces




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Our Results


Summary

The Effect of the Oblateness of the Central Body

The Impact of the Earths J on the Lense-Thirring Effect


Outline







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Summary




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The Impact of the Even Zonal Harmonics on the Orbit

of a Test Particle

A rotating body is oblate. The even zonal harmoniccoefficients

J, = 2,4, . . . of the multipolar expansionof the Newtonian

gravitational potential of the central body induce secularprecessionson and as well [Kaula 1966] much largerthanthe LT ones. For = 2 the node precession is

J2 =

3

2nR

a

2cos i J2

(1 e2)2

R equatorial radius of the central body of mass M

n = GM/a3 mean motion of the test particleL. Iorio Attempts to Measure Frame-Dragging in the Solar System



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Reducing the Impact of the Earths Even Zonals

The J are estimated as least-square fit-for parameters of

global solutions of the Earth gravity field by processing

huge multi-year data records of dedicated spacecraft like

GRACE. Thus, their knowledge is imperfect. The node/pericentre precessions due to our bad

knowledge J of the even zonals induce systematicuncertaintyin the LT signal.

To reduce such a bias, suitable linear combinations[Ciufolini 1996] of the nodes of existing (LAGEOS,

LAGEOS II) [Ries et al. 2003, Iorio and Morea 2004] and

future (LARES) [Iorio 2005a] satellites have been proposed

and used; they cancel out some low-degree even zonals.




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g


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Assessment of the Systematic Bias due to J

LAGEOS and LAGEOS II orbit at about 6000 km; they are

sensitive to just a fewJ of lowdegrees. LARES should fly

at about 1450 km only; it will be affected by moreJ of

higherdegrees.

The LAGEOS-LAGEOS II node combination

[Iorio and Morea 2004] used so far

[Ciufolini and Pavlis 2004] cancels out J2 and is affected by

J4, J6, J8, . . .. The LAGEOS-LAGEOS II-LARES node

combination [Iorio 2005a] to be used cancels out J2, J4and is affected by J6, J8, . . ..

Many Earths gravity field models are nowadays available.

If one uses for J their covariance sigmasJ one getssystematic errors 5 10% (LAGEOS-LAGEOS II).



Th S i E f G i i l O i i Th Eff f h Obl f h C l B d


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higherdegrees.










Th S t ti E f G it ti l O i i Th Eff t f th Obl t f th C t l B d


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higherdegrees.












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The Lense-Thirring EffectThe Systematic Error of Gravitational Origin


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What Is the Real Uncertainty in J, = 2, 4, . . . ?

A more conservative approach consists of taking the em

differences

[Lerch et al. 1994, Ciufolini 1996, Lucchesi 2007]

J

= J(X) J

(Y) between the estimated values for

different pairs of models X and Y and see if they are larger

than the combined sigmas = J(X) + J(Y) of thatmodels. It turns out that J > for manymodels just forthose even zonals which are most relevant for us.

The GRACE-based models considered here, by fivedifferent institutions, are AIUB-GRACE01S (AIUB),

EIGEN-GRACE02S (GFZ), GGM02S (CSR), GGM03S

(CSR), ITG-Grace02s (IGG), ITG-Grace03s (IGG),

JEM01-RL03B (JPL).




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Summary



What Is the Real Uncertainty in J, = 2, 4, . . . ?

A more conservative approach consists of taking the em

differences

[Lerch et al. 1994, Ciufolini 1996, Lucchesi 2007]

J

= J(X) J

(Y) between the estimated values for

different pairs of models X and Y and see if they are larger

than the combined sigmas = J(X) + J(Y) of thatmodels. It turns out that J > for manymodels just forthose even zonals which are most relevant for us.

The GRACE-based models considered here, by fivedifferent institutions, are AIUB-GRACE01S (AIUB),

EIGEN-GRACE02S (GFZ), GGM02S (CSR), GGM03S

(CSR), ITG-Grace02s (IGG), ITG-Grace03s (IGG),

JEM01-RL03B (JPL).




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Models compared (LAGEOS-LAGEOS II) AIUB-GRACE01SJEM01-RL03B 20%

AIUB-GRACE01SGGM02S 27%AIUB-GRACE01SEIGEN-GRACE02S 33%

JEM01-RL03B

GGM03S 17%JEM01-RL03BITG-Grace02 18%

JEM01-RL03BITG-Grace03s 20%GGM02SGGM03S 24%

GGM02SITG-Grace02 25%

GGM02SITG-Grace03s 27%GGM03SEIGEN-GRACE02S 30%

ITG-Grace02EIGEN-GRACE02S 31%ITG-Grace03sEIGEN-GRACE02S 33%



A M C ti A h


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(Models compared = 60; L-LII-LR) (SAV) (RSS)

AIUB-GRACE01SJEM01-RL03B 23% 16%AIUB-GRACE01SGGM03S 22% 13%AIUB-GRACE01SITG-Grace02 24% 15%

AIUB-GRACE01SITG-Grace03 22% 14%AIUB-GRACE01SEIGEN-GRACE02S 14% 7%JEM01-RL03BGGM02S 14% 9%JEM01-RL03BEIGEN-GRACE02S 26% 15%GGM02SITG-Grace02 16% 8%

GGM03SEIGEN-GRACE02S 24% 13%ITG-Grace02EIGEN-GRACE02S 25% 14%ITG-Grace03sEIGEN-GRACE02S 24% 13%



A More Conser ati e Approach


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y g

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Outline








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y g

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More Conservative Evaluations of the Systematic

Gravitational Errors in the SLR Lense-Thirring Tests

With J for the uncertainty J the systematic error in theLAGEOS-LAGEOS II test is 25 37% [Iorio 2009].

The LAGEOS-LAGEOS II-LARES test is complicated bythe higher degreeseven zonals brought into play by

LARES. A computation up to = 60 with the standardKaulaapproach [Kaula 1966] and J yields 5 30%[Iorio 2009]. Another calculation up to = 70 with Jyields even larger figures.

The atmospheric (neutral and charged) drag on LARES

makes its inclination secularly decreasing; this yields a

node bias J2 of 3 9% yr1 [Iorio 2008b].





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O R l


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Suns Gravitomagnetic Length and Orbit Accuracy

The characteristic gravitomagnetic length for the Sun is

S

Mc= 319 m

The (formal) uncertainty in the mean orbit radius of the inner

planets is, according to E.V. Pitjeva who used the EPM

ephemerides:

Mercury Venus Earth Mars

38 m 3 m 1 m 2 m

Thus, it seems possible to measure the Suns LT



O R lt


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Our Results


Summary

The Perihelion Precession of Venus

E.V. Pitjeva has recently fitted almost one century of planetary

data of all kinds, including also ranging data from Magellan,

with the EPM2008 ephemerides. She estimated a correction to

the standard Newtonian/Einsteinian perihelion precession

Venus = 0.0004 0.0001 arcsec cy1

where the error quoted is formal; the realisticone may be up to

5 times larger. accounts, by construction, for all

unmodelled/mismodelled effects: LT was not modelled. Thepredicted LT perihelion precession of Venus is

(LT)Venus = 0.0003 arcsec cy

1



Our Results


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Estimated Perihelion Extra-Precessions

Table: First line: corrections , in 104 arcsec cy1 , to thestandard Newton/Einstein perihelion precessions of the inner planetsestimated by E.V. Pitjeva with the EPM ephemerides. The result for

Venus has been obtained by recently processing radiometric datafrom Magellan spacecraft. The errors in square brackets are theformal ones. Second line: LT predicted precessions LT, in 10

4

arcsec cy1 .

Mercury Venus Earth Mars

36 50 [42] 4 5 [1] 2 4 [1] 1 5 [1]

LT 20 3 1 O(105)



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Summary

Compatibility of the Solar LT Precessions with the

Estimated Ones

Table 1 shows that the predicted LT perihelion precessions LT

are compatiblewith the estimated corrections to theNewtonian/Einsteinian precessions for all the innerplanets.

Moreover, the ratios AB A/ B are equal to the

predicted LT ratios

LAB

(LT)

A/

(LT)

B= a

B(1 e2

B)3/2/a

A(1 e2

A)3/2, within the

errors, for all the pairsof inner planets. Many long-range

models of modified gravity do notpass such a test.



Our Results


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Our Results


Summary

Sources of Systematic Errors in the Solar LT tests

The main source of systematic uncertainty is represented,

also in this case, by the mismodelling in the first even zonal

harmonic J2 of the Sun, which is presently known with an

uncertainty of about 10% with respect to its nominal valueJ2 = 2 10

7. The resulting mismodelled perihelion

precession of Venus amounts to +0.0002 arcsec cy1

The impact of the mismodelling in the asteroid ringamounts to +0.0007 0.0001 arcsec cy1 for Venus



Our Results


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Our Results


Summary

Sources of Systematic Errors in the Solar LT tests

The main source of systematic uncertainty is represented,

also in this case, by the mismodelling in the first even zonal

harmonic J2 of the Sun, which is presently known with an

uncertainty of about 10% with respect to its nominal valueJ2 = 2 10

7. The resulting mismodelled perihelion

precession of Venus amounts to +0.0002 arcsec cy1

The impact of the mismodelling in the asteroid ringamounts to +0.0007 0.0001 arcsec cy1 for Venus



Our Results


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Summary

Combining Planetary Perihelia and Nodes

The linear combination approach used for the LAGEOS

satellites can be also applied to the various planets

perihelia to cancel out selected sources of systematic

errors [Iorio 2005b]

In principle, also the corrections to the standardNewtonian node precessions can be used to measure the

Suns LT, although, at present, they have not yet beenestimated [Iorio 2005b]



Our Results


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Summary

Combining Planetary Perihelia and Nodes

The linear combination approach used for the LAGEOS

satellites can be also applied to the various planets

perihelia to cancel out selected sources of systematic

errors [Iorio 2005b]

In principle, also the corrections to the standardNewtonian node precessions can be used to measure the

Suns LT, although, at present, they have not yet beenestimated [Iorio 2005b]



Our Results


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Summary

Mars Global Surveyor and Frame Dragging

The time series of the Root Mean Square (RMS) orbit overlap

differences [Konopliv et al. 2006] of the out-of-plane part ofthe orbit of the Martian polar spacecraft Mars Global Surveyor(MGS) over a time span P 5 yr, processed withoutmodelling gravitomagnetism at all, has been successfully

interpreted in terms of LT [Iorio 2006]. Criticisms have been

raised in [Krogh 2007]; the reply is in [Iorio 2007]



Our Results


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Summary

Two Independent Tests of Frame Dragging with MGS

The average of the time series of the out-of-planeRMS orbit overlap differences, normalized to the average

LT shift LT over the same time span, is

= 1.002 0.005 Linear fits of the time series for different time spans, i.e.

including different sets of data points, yield results in

agreement with the LT interpretation. For example, the fit

of the entire data set yields a slope of 1.03 0.42, inLT-normalized units; removing the data for the monthJanuary 2001, likely affected by major measurement

errors, yields a normalized slope 0.98 0.42



Our Results


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Summary

Two Independent Tests of Frame Dragging with MGS

The average of the time series of the out-of-planeRMS orbit overlap differences, normalized to the average

LT shift LT over the same time span, is

= 1.002 0.005 Linear fits of the time series for different time spans, i.e.

including different sets of data points, yield results in

agreement with the LT interpretation. For example, the fit

of the entire data set yields a slope of 1.03 0.42, inLT-normalized units; removing the data for the monthJanuary 2001, likely affected by major measurement

errors, yields a normalized slope 0.98 0.42



Our Results

S l d M i F D i d i M bili


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Summary

Reply to Some Criticisms

The exceedingly large gravitational bias due to the

mismodelling in the areopotential which should plague the

LT signal according to Krogh are, in fact, absent from the

data, both if one refers to the average over the full,

multi-year time span and if a single arc 6-days long is,instead, considered

The non-gravitational perturbations like, e.g., the

atmospheric drag, mainly affect the in-plane orbit

components, i.e. the radialand transverseones, not theout-of-plane one, affected by LT. Indeed, advances in

modelling the non-gravitational forces improved just the

in-plane components, leaving, instead, the out-of-plane

one substantially unaffected



Our Results

S l d M ti F D i d it M bilit


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Summary

Reply to Some Criticisms

The exceedingly large gravitational bias due to the

mismodelling in the areopotential which should plague the

LT signal according to Krogh are, in fact, absent from the

data, both if one refers to the average over the full,

multi-year time span and if a single arc 6-days long is,instead, considered

The non-gravitational perturbations like, e.g., the

atmospheric drag, mainly affect the in-plane orbit

components, i.e. the radialand transverseones, not theout-of-plane one, affected by LT. Indeed, advances in

modelling the non-gravitational forces improved just the

in-plane components, leaving, instead, the out-of-plane

one substantially unaffected



Our Results

Solar and Martian Frame Dragging and its Measurability


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Summary

Summary

A conservative evaluation of in the LAGEOS-LAGEOS IItest yields 25 37%.

The low altitude of LARES makes more even zonals

relevant; their uncertainty yields for the LAGEOS-LAGEOSII-LARES 5 30%, or even larger.

The secular decrease of the LARES inclination due to the

atmospheric drag yields a bias 3 9% yr1.

Measuring Suns LT using the motions of the inner planetsseem to be feasible in the near future.

A preliminary positive test of LT with the MGS spacecraft in

the field of Mars has been reported



Our Results



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Appendix References


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References I

L. Iorio (ed.)

The Measurement of Gravitomagnetism: A Challenging

Enterprise.

NOVA, Hauppauge (NY), 2007.W.M. Kaula

Theory of Satellite Geodesy.

Blaisdell, Waltham, 1966.

I. CiufoliniN. Cim. A 109(12), 1709-1720, 1996.

I. Ciufolini, and E.C. Pavlis

Nature431, 958-960, 2004.


Appendix References


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References II

L. Iorio.

New Astron. 10(8), 616-635, 2005a.

L. Iorio.

Astron. Astrophys. 431(1), 385-389, 2005b.

L. Iorio.

Class. Quantum Grav. 23(17), 5451-5454, 2006.

L. Iorio.

J. Gravit. Phys. , at press, gr-qc/0701146v5

L. Iorio.

Schol. Res. Exch. 2008, ID 105235, 2008.


Appendix References


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References III

L. Iorio.

arXiv:0809.3564v1 [gr-qc]

L. Iorio.

Space Sci. Rev. doi:10.1007/s11214-008-9478-1 2009.

L. Iorio, and A. Morea.

Gen. Relativ. Gravit. 36(6), 1321-1333, 2004.

A. Konopliv et al.

Icarus182(1), 23-50, 2006.

K. Krogh

Class. Quantum Grav. 24(22), 5709-5715, 2007.


Appendix References


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References IV

F.J. Lerch, et al.

J. Geophys. Res. 99(B2), 2815-2839, 1994.

D.M. Lucchesi

Adv. Sp. Res. 39(10), 1559-1575, 2007.

J.C. Ries, et al.

In: R.J. Ruffini, C. Sigismondi (eds.) Nonlinear

Gravitodynamics, pp. 201-211. World Scientific, Singapore,2003.



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Acknowledgements

I gratefully thank the organizers of this wonderful School for

their kind invitation and for their warm hospitality!


l. iorio- recent attempts to measure the general relativistic lense-thirring effect with natural and...

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