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On the Relativistic Theory of Earth Rotation
S.A. Klioner

Lohrmann Observatory, Dresden Technical University

JD6, XXVII GA of the IAU, Rio de Janeiro, Brasil, 6 August 2009
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Relativity and Earth rotation: why to bother?
· Earth rotation is the only astronomical phenomenon - which is observed with a high accuracy and - which has no widely-used consistent relativistic model

· Modern theories of precession/nutation (IAU2000) are based on purely Newtonian theories with geodetic precession and nutation added in an inconsistent way · Modern theories of rigid Earth nutation are intended to attain formal accuracy of 1 µas (expected relativistic effects are much larger)
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How to model?
· Early attempts ( - 1986 ) - One single reference system BCRS for both translational motion of solar system and for rotational motion of all the bodies... - The results were clearly physically inadequate coming from bad choice of coordinates: E.g. spurious annual variations in LOD with an amplitude of 75 µs... Reason: "bad" coordinates that provide no analogy of Newtonian tidal forces at the post-Newtonian level "Better" coordinates are clearly needed: GCRS

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How to model?
· More sophisticated way (1986 - ) - A physically adequate local GCRS - Still some coordinates, but chosen in such a way that the influence of external gravitational fields is as small as possible: full analogy of Newtonian tidal forces at the post-Newtonian level

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Main goal of the project
· Derivation of a new consistent and improved precession/nutation series for a rigidly rotating multipole model of the Earth in the post-Newtonian approximation of general relativity · using post-Newtonian definitions of : - potential coefficients - moment of inertia tensor · dynamical equations in the GCRS · correct relativistic time scales · rigorous treatment of the geodetic precession and nutation


Equations of rotational motion in the GCRS
- Post-Newtonian equations of rotational motion in the GCRS (Damour, Soffel, Xu, 1993, Klioner, Soffel et al 1996-)

d C ab dTCG

(

b

)
=
l =1

1 l!

abc

M bLGcL + La (C, ,

iner

) + ...

The last terms is the Coriolis torque from the relativistic precessions:



a iner

3 2 i 1 i i = 2 aij v E j wext ( x E ) + 2 aij j wext ( x E ) 2 aibv E Gb 2c x c x 2c

geodetic precession Lense-Thirring precession
wext ( x ) =
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A E



GM

A A

xx

, weixt ( x ) =

A E



GM

A A

xx

v

i A

Thomas precession (negligible)


Rigidly rotating multipoles in the GCRS
· Klioner, Soffel, Xu, Wu, 2001 (based on many previous results): - Post-Newtonian equations of rotational motion in the GCRS

d C abb = dTCG

(

)
l =1

1 l!

abc

M bLGcL + La (C, ,

iner

) + ...

- Rigidly rotating multipoles: several assumptions on the multipole moments and the tensor of inertia

C ab = P ac P bd C ab , C M
a a1 a2 ... al

ab

= const,
al bl

=P

a1 b1

P

a2 b2

...P

M

b1 b2 ...bl

,M

b1 b2 ...bl

= const, l 2,

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1 d db = abc P P 2 dTCG P ab (TCG ) is an orthogonal

dc

matrix defining the orientation of the ITRS in GCRS


Numerical code: an overview
· Fortran 95, about 20000 lines · careful coding to avoid excessive numerical errors · two numerical integrators: ODEX and ABM with dense output · automatic accuracy check: forth and back integrations · any available arithmetic: 64 bit, 80 bit, 128 bit · extended-precision arithmetic for precision-critical operations (switchable) · the STF code has been automatically generated by Mathematica · baseline: ODEX with 80 bits on Intel architecture gives errors <0.001 µas for 150 years · in the Newtonian limit reproduces SMART within the errors of the latter · performance: Newtonian case: all the relativity on:
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2.2 sec per yr 8.8 sec per yr


Long-term numerical integrations
A first step to a relativistic theory of precession ...

A long solar system ephemeris is needed: DE404 is used to check the situation: 6000 yr

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Noise from the downgrade: DE404 vs DE403

µas

10

years from J2000


Effects of the post-Newtonian torque
6000 years with DE404

µas

11

years from J2000


Effects of the post-Newtonian torque
6000 years with DE404:

= -146.67 + 640.60 t - 7921.70 t + 11375.50 t + 1308.23t
2 3 4

2

= -0.61 - 1560.31t + 3.22 t 2 + 2.13t 3 - 0.06 t = -0.23 + 0.014 t - 7.99 t + 0.64 t + 0.08 t
in µas, t is in thousand years
3 4

4

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Effects of the post-Newtonian torque

6000 years with DE404: minus 4th-order polynomial

µas

13

years from J2000


Effects on the LOD
6000 years with DE404:

+ cos =
rad/s

years from J2000

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Effects on the LOD
6000 years with DE404:

+ cos =
rad/s

years from J2000

N < 450 µas / d LODN < 30 µs
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Effects on the LOD
6000 years with DE404:

+ cos =
rad/s

years from J2000


16

pN

-

Newt

< 0.8 µas / d LOD

pN

< 0.06 µs