Comparisons of precession-nutation models - IVS-VLBI

Comparisons of precession-nutation models

N. Capitaine(1), P.M. Mathews(2), V. Dehant (3), P.T. Wallace(4), S. Lambert (1) (1) Observatoire de Paris-SYRTE, France, (2) University of Madras, India, (3) Royal Observatory of Belgium, Belgium, (4) Rutherford Appleton Laboratory, UK

IVS General Meeting, St-Petersburg, March 2008

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Presentation prompted by recent discussion about a comments paper by Mathews, Capitaine, Dehant (astro-ph 0710.0166) on the ERA2005 model (see Krasinski, Krasinski & Vasilyev CMDA 2006)

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Aim: to clarify the comparison between different precession-nutation models (especially IAU 2000/2006 and ERA 2005)

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Issues –

Key characteristics of the IAU 2000/2006 precession-nutation models

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Comparison of the main features of the IAU and ERA models : comments

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Comparison of the IAU models with VLBI observations

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Comparison of P03 with INPOP06 numerical integration


Key characteristics of the IAU 2000/2006 models


IAU Precession-nutation 1) IAU 2000 Resolution B1.6 - adopted the IAU2000 precession-nutation (Mathews, Herring, Buffett 2002) which

was implemented in the IERS Conventions 2003 IAU 2000A Nutation (non-rigid Earth)

IAU 2000 Precession= IAU 1976 (Lieske et al. 1977) + corrections to precession rates 1st step dψΑ (IAU 2000) = −0.299 65”/c ; dωΑ (IAU 2000) = −0.025 24”/c Celestial pole offsets at J2000 (VLBI estimates) ξ0 (IAU 2000) = −16.6170 mas ; η0 (IAU 2000) = −6.8192 mas

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recommended the development of new expressions for precession consistent with dynamical theories and with IAU 2000A nutation

2) IAU 2006 Resolution B1 -

adopted the P03 precession (Capitaine, Chapront, Wallace, 2003) dynamical model 2nd step consistent with IAU 2000A nutation and with non-rigid Earth recommended improved definitions (ecliptic, precession of the equator, precession of the ecliptic) IVS General Meeting, St-Petersburg, March 2008

IAU 2000 nutation Rigid Earth nutation

→

MHB2000 nutation

transfer function (Mathews et al. 2002) analytical solution (Souchay et al. 1999) semi-analytical series: 1365 luni-solar and planetary terms Transfer function derived from the solution of equations obtained by generalization of the SOS equations (Sasao et al. 1980) with Basic Earth Parameters fitted to VLBI data.

SOS equations

Basic Earth Parameters ef κ γ e Im K(CMB) Re K(ICB) Im K(ICB) rms residuals

Estimate 0.0026456 ±20 0.0010340 ±92 0.0019662 ±14 0.0032845479 ±12 - 0.0000185 ±14 0.00111 ±10 - 0.00078 ±13 0.0132 mas

+ obliquity rate fitted to VLBI


Correction to

hydrostatic equilibrium

0.0000973 - 0.0000043 0.0000007 0.000037

IAU 2000A luni-solar nutations

Table 5.3a (IERS Conventions 2003)

L Lm

F

D Om

Period (days)

* * 0 0 0 0 0 0 1 0

0 0 0 0 1 1 0 0

0 0 2 -2 2 0 0 0 0 0 2 -2 0 0 2 0

1 2 2 2 0 2 0 1

-6798.383 182.621 13.661 -3399.192 365.260 121.749 27.555 13.633

Psi (mas) -17206.4161 -1317.0906 -227.6413 207.4554 147.5877 -51.6821 71.1159 -38.7298

In Phase Out of phase dPsi/dt Eps dEps/dt Psi dPsi/dt Eps Deps/dt (mas/c) (mas) (mas/c) (mas) (mas/c) (mas) (mas/c) -17.4666 9205.2331 0.9086 3.3386 0.0029 1.5377 0.0002 -0.1675 573.0336 -0.3015 -1.3696 0.0012 -0.4587 -0.0003 -0.0234 97.8459 -0.0485 0.2796 0.0002 0.1374 -0.0001 0.0207 -89.7492 0.0470 -0.0698 0.0000 -0.0291 0.0000 -0.3633 7.3871 -0.0184 1.1817 -0.0015 -0.1924 0.0005 0.1226 22.4386 -0.0677 -0.0524 0.0002 -0.0174 0.0000 0.0073 -0.6750 0.0000 -0.0872 0.0000 0.0358 0.0000 -0.0367 20.0728 0.0018 0.0380 0.0001 0.0318 0.0000

….. semi-analytical series with “in-phase” and “out-of phase” components (1365 luni-solar and planetary terms; amplitudes between 17.2” and 0.1 µas ; periods between 3d and 101 cy)


P03 precession expressions for the basic quantities

mas

mas/cy

mas/cy2

mas/cy3

mas/cy4

mas/cy5

Ecliptic Equator

(Capitaine et al. 2003)


The P03 precession of the equator Precession equations (Williams, 1994): Contributions to the precession rates rψ , rε

(r0)1 and (u0)1 derived from VLBI observations + correction in the precession rate for the change in the J2000 obliquity from IAU2000 to P03

Contributions to the precession rates rψ , rε from Williams 1994, Brumberg et al. 1998, Mathews et al. 2002 IVS General Meeting, St-Petersburg, March 2008

Variations in the Earth’s oblateness during the past 28 years (from Cheng & Tapley; JGR109, B09402, 2004) secular decrease dJ2/dt ~ − 2.8 x10-11/ yr i.e. de/dt ~ − 8.5 x10-11/ yr

- seasonal annual variations (~ 2.9 x10-10) - interannual variations ~ 4-6 years (related to strong El Niño-SO events) - variation of period of 21 yr, amplitude of 1.4 x10-10

Cheng & Tapley 2004

Cox and Chao 2002

Note that: - the J2 rate effect on precession has been taken into account in the IAU 2006 precession, with the value dJ2/dt = − 3.0 x10-11/ yr (i.e. de/dt = − 9.1 x10-11/ yr) - the effect of the observed variations in J2 on precession has been discussed by Bourda & Capitaine (2004) - the variations in J2 (or e) cannot be perfectly predicted over periods of 10 years IVS General Meeting, St-Petersburg, March 2008

Comparison of semi-analytical precession expressions

differences less than 1 mas/cy2 1 mas/cy3 etc. except for the J2 rate contribution

B03: Bretagnon et al. 2003 F03: Fukushima 2003 P03 Capitaine et al. 2003


The numerical ERA model in comparison with the semi-analytical IAU models : comments


Centrifugal perturbation and Love number formalism •

MHB 2000: based on the SOS equations of Sasao-Okubo-Saito (1980) for the variations in rotation of the Earth’s mantle and fluid core. – –

Hd is a key factor for the precession rate and the nutation amplitudes

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using the ERA value SERA for the scale factor would give a difference in the precession rate of -0.14’’/cy (i.e. 0.4 mas in the 18.6-yr nutation)

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these equations lead to the scale factor SMHB=Hd=e/(1+e) for the precession rate the tidal contribution to the centrifugal potential is only the incremental part arising from the time dependent dΩ generated by the action of the perturbing potential the Earth’s deformational response to constant forcing over very long periods is very far from that of an elastic solid (e.g. mantle convection) is characterized by the fluid/secular Love number kf=ks~0.9~3 k2, appropriate to a fluid-like behaviour hydrostatic equilibrium structure of the axially symmetric ellipsoidal Earth (Clairaut ‘s equation). This structure leads to observed value for e=(C-A)/A

ERA 2005: based on « revised SOS equations » – – – –

treats the centrifugal potential due to the full angular velocity vector Ω = Ω0 + dΩ as a « tidal » perturbation (though Ω0 is quite unrelated to tidal potentials) the Earth’s deformational response to constant forcing is just like its elastic response to periodic forcing at tidal frequencies; the Love number k representing the response has the same value k2~0.3 in both cases the revised SOS equations are based on the above new concepts leads to the scale factor SERA=e/(1+ e − eσ/3), with σ= k2/ks ~0.3, for the precession rate. Dependence on deformability (through k2) is a radical change


Modeling of the dissipative phenomena •

MHB 2000: spells out the mechanisms which produce the out-of-phase parts of nutation amplitudes and models them on the basis of actual observational data on the various phenomena involved. –

– –

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mantle analesticity (model employed by Wahr and Bergen, deduced from seismological observations); model is incorporated into the expressions for the increments of inertia of the whole Earth and the fluid core ocean tide heights from CSR4 tables based on space-geodetic observations core-mantle and outer core to inner core couplings due to the magnetic fields at the boundaries, modelled from theory; complex coupling constants KCMB and KICB

ERA 2005: no specific mechanisms are invoked for the dissipation. –

model for – dissipation is essential for computing the out-of-phase nutations

models dissipation in the Earth in terms of three phase delay parameters δ, δc, δi for the deformational responses in the mantle and the outer and inner fluid cores to tidal perturbations, plus the dissipative part kdis of a CMB coupling the ocean tide effects in nutations are effected through an increment dk2 to the Love number k2 , not constrained by any OT observations : dk2= k2 (1) cos θB + k2 (2) cos θB → d k2 depends only on the colatitude θB of the tide producing body in the sky at that instant and has identical values at all points of the globe at any instant (*)(**) (*) seems

very unphysical because the ocean tide is highly non-uniform over the Earth’s surface (**) in a similar-looking formula of Kaula (1969), the dependence is on the co-latitude of the observation site, which makes sense physically


Earth model •

MHB 2000 –

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• model for the dynamics of the Earth’s interior is essential for computing the NRE nutations

spherically symmetric classical models used of the radial variation of the properties of a nonrotating Earth, based on seismological data (e.g. PREM) models of the ellipsoidal structure of the interior of the steadily rotating Earth are derived from such models using hydrostatic equilibrium theory (Clairaut’s theory)

ERA 2005 –

rejects the « classical » Earth models

(even the non-rotating Earth has to have ellipticity ~(2/3)e, as the centrifugal potential of the steadily rotating Earth can account for only ~(1/3)e as ERA theory takes k to be k2 (instead of ks) even under constant forcing)

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proposes no mechanism to determine the detailed interior structure → large deviations of the ERA estimates for some of the Earth parameters from values computed on the basis of seismological Earth models (e.g. value for ef higher than e) model postulates a fluid inner core region which has the same physical properties as, and is nevertheless dynamically distinct from, the fluid outer core


Estimation of parameters •

MHB 2000 – – – –

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7 parameters were treated as adjustable and employed for fitting the theoretical outputs to the VLBI the chosen parameters were the 4 Earth parameters e, ef, κ, γ, and 3 of the 4 real/imaginary parts of KCMB and KICB. These parameters influence the nutation amplitudes independently of each other, and to a sufficient extent so that their estimation may be done readily minor tweaking of the input values of 1 of the 2 parameters in the anelasticity model and 1 parameter in the ocean tide model was done at the end; and the real and imaginary parts of the prograde annual amplitude were taken as adjustable obliquity rate fitted to VLBI

ERA 2005 –

Estimating a large number of parameters does reduce the RMS of the residuals, but highly correlated parameters make the solution not reliable

the estimated parameters are: • e and k2 , • ec, k2c, k2v , α : high mutual correlations • νv, kel , kdis (which do not appear in any of the resonance frequencies) • 4 other « geophysical » parameters, together with 6 « empirical » parameters (e.g. E1 in order to correct the ratio of the retrograde and prograde nutation amplitudes and E2 to correct the out of phase nutations) • the FCN frequency as obtained from prior fits by others is used as an input constraint while estimating ERA parameters.


Comparisons of the IAU models with VLBI observations


X CIP residuals

X CIP coordinate

top: IAU2000A − VLBI (IAA) bottom: (P03 + IAU2000A nut) − VLBI (IAA) without removal of an FCN model


X and Y CIP residuals

P03 precession+IAU2000A nutation −VLBI (IAA)

top: X bottom: Y after removal of the FCN (time-varying sinusoidal with a constant period of P = −430.23 days)

A curvature is visible both in X and Y with approximately the same amplitude and the same phase


ivs07q4e Original series of the VLBI residuals w.r.t. IAU 2000A FCN removed

FCN + degree 2 polynomial removed


Residuals w.r.t. IAU 2000A: various smoothed VLBI series

result of a 2-yr moving average on the time series after the FCN is removed

similar curvature for all the series


Various fits of a parabola in the VLBI residuals Differences between data centres t

t2

rms

Correlations 1 t t -0.3 t2 -0.7 +0.3

differences up to: 400 µas/cy in X, 3.8 mas/cy2 in Y

Differences between the models t

t2

rms

P03 versus IAU 2000: discrepancies larger by −3 mas/cy2 in X (J2 rate effect), lower by −500 µas/cy in Y IVS General Meeting, St-Petersburg, March 2008

Various fits of a parabola in the VLBI residuals

IVS combined solution: Differences between various IVS procedures t

IAU 2000 based IAU 1976/80 based

differences up to: 4.2 mas/cy2 in X, 5.7 mas/cy2 in Y


t2

rms

The effect of the celestial frame instability Two different solutions: (i) all sources as global parameters, (ii) all sources as session parameters Nutation offsets corrected from ‘local’ rotations (= rotation angles between ‘local’ celestial reference frames and ‘global’ one) ‘Local’ rotations of the CRF

Nutation offsets: from ‘global’ solution (red), and from corrected ‘session’ solution (blue)

same curvature of the residuals for (i) and (ii) IVS General Meeting, St-Petersburg, March 2008

Comparisons of the P03 precession with the numerical planetary ephemeris INPOP06


Comparison over 5000 years of the P03 precession with the numerical planetary ephemeris INPOP06 (from Fienga et al., 2008, A&A 477, 315)

- good agreement up to 1000 yr - discrepancies after 1000 yr due to the polynomial representation of the semi-analytical precession


Comparison over 400 years of P03 + IAU2000A with the numerical planetary ephemeris INPOP06

(axis of angular momentum, external torque modified by the non-rigid Earth)

Conclusion: no quadratic differences between INPOP06 and P03

Credit: INPOP team (2008)

- based on totally independent models (except for the linear term in X) - totally different solutions


Comparison plots in the time domain


ERA2000 – IAU2000 comparisons

current observed residuals

ERA 2005 predictions for VLBI residuals w.r.t. IAU2000 (Krasinski 2006) same scale as above


Concluding remarks -

The key characteristics of the IAU 2000/2006 precession-nutation models have been reviewed

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No significant discrepancies appear between the IAU models and other semi-analytical solutions

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Comparison of the P03 precession with INPOP06 numerical integration shows a very good agreement over 2 centuries

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The comments of Mathews et al. 2007 (astro-ph 0710.0166) about the ERA model have been reaffirmed

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Comparison of the IAU models with VLBI observations show an unexplained curvature: quadratic terms do not seem physically possible → long-periodic term? spurious effect of the analyses?

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The celestial motion of the CIP may not be totally predictable, which shows that the IERS celestial offsets are still necessary (e.g. the effect on precession due to the variations in J2 cannot be perfectly predicted)
