A&A 432, 355-367 (2005)
DOI: 10.1051/0004-6361:20041908

Improvement of the IAU 2000 precession model

N. Capitaine¹ - P. T. Wallace² - J. Chapront ¹

1 - Observatoire de Paris, SYRTE/UMR 8630 - CNRS, 61 avenue de l'Observatoire, 75014 Paris, France
2 - H. M. Nautical Almanac Office, Space Science and Technology Department, CLRC / Rutherford Appleton Laboratory, UK

Received 26 August 2004 / Accepted 3 November 2004

Abstract
The IAU 2000 precession consists of the IAU 1976 ecliptic precession (Lieske et al. 1977, A&A, 58, 1) and the precession part of the IAU 2000A equator adopted by IAU 2000 Resolution B1.6 (Mathews et al. 2002, J. Geophys. Res., 107, B4, 10.1029/2001JB000390). In this paper we provide a range of new expressions as possible replacements for the IAU 2000 precession. The new expressions are based upon the so-called P03 solution of Capitaine et al. (2003b, A&A, 412, 567) for the equator and the ecliptic. In addition an improved model for the precession of the equator is discussed. This improved solution was obtained in exactly the same way as P03 but using a refined model for the contributions of the non-rigid Earth (Mathews 2004, private communication) and revised integration constants for the precession rates resulting from fits to the most recent VLBI data. The paper reports on the procedure that was used for improving the P03 solution and on the comparisons of this solution with the MHB 2000, IAU 2000 and P03 solutions. It also discusses the choices for the solution to be put forward as a replacement for IAU 2000. We concluded that the existing VLBI data were insufficient to provide convincing evidence that the improved solutions would deliver better accuracy than the existing P03 solution, and we recommend retaining P03 as the replacement for IAU 2000. P03, which unlike the IAU 2000 precession is dynamically consistent, has the advantage of already having been used experimentally by a number of groups; the model is recalled in Tables 3-5. Due to the strong dependence of the precession expressions on the precession rates and of the precession in longitude (or equivalently the celestial CIP X coordinate) on the J₂ rate model, we also provide a parameterized P04 solution for these quantities as functions of those parameters. The expressions include the quantities to be used in both the equinox-based and CIO-based (i.e. referred to the Celestial Intermediate Origin) transformations.

Key words: astrometry - reference systems - ephemerides - time

   
1 Introduction

The current IAU precession that is implemented in the IERS Conventions 2003 consists of (i) the IAU 1976 precession (Lieske et al. 1977) and (ii) the IAU 2000A precession components (Mathews et al. 2002, denoted MHB in the following) adopted by Resolution B1.6., which are merely corrections to the precession rates in longitude and obliquity of the IAU 1976 precession. The IAU 2000 precession, although of practical utility for the next few years, is in fact dynamically inconsistent and suffers, except for the improvements in the precession rates, from the same limitations as the IAU 1976 precession in the precision of the coefficients and compliance with up to date models for the ecliptic motion. As recommended by IAU Resolution B1.6, new precession models consistent with IAU 2000A have been studied, and this work has been reported in recent papers (Bretagnon et al. 2003; Fukushima 2003; Capitaine et al. 2003b). An IAU Division I Working Group on "precession and the ecliptic'' was established at the 2003 General Assembly in order to look into these issues and select an improved, dynamically consistent, precession model to be proposed as a replacement for the precession component of IAU 2000A.

The solution by Capitaine et al. (2003b) (denoted P03 in the following) made a clear distinction between (i) the precession of the ecliptic due to planetary perturbations and (ii) the precession of the equator due to the luni-solar and planetary torques on the oblate Earth, both motions being expressed with respect to inertial space.

The P03 ecliptic precession was derived using both the analytical solution VSOP87 (Bretagnon & Francou 1988), which modeled the periodic terms, and the JPL numerical ephemeris DE406 (Standish 1998) fitted over a 2000-yr interval, which provided the secular portion of the model. Capitaine et al. (2004) tested the various 2003 solutions against the best numerical ephemerides and, after investigating the deficiencies in the other models, concluded that P03 was the most accurate ecliptic model currently available. This solution (the accuracy of which is estimated to be about 0.05 mas/cy over a two-millennium interval centered on J2000) is thus proposed as a replacement for the current IAU 1976 model (Lieske et al. 1977) for the precession of the ecliptic.

The P03 equator precession solution was obtained by solving the dynamical precession equations, with (i) the P03 ecliptic precession, (ii) theoretical contributions to precession rates for a non-rigid Earth model from Williams (1994) and Mathews et al. (2002) and (iii) integration constants that were derived from the MHB estimates of precession rates by applying corrections for perturbing effects on the observed quantities (Capitaine et al. 2003b). One of these effects is due to the fact that the "observed'' quantity for the precession in longitude has been shown to be not the precession in longitude itself, as generally considered, but instead its projection parallel to the equator on the conventional ecliptic with the obliquity used in the VLBI software (i.e. the IAU 1976 value) resulting in a dependence of the "observed'' quantity for a given model on the obliquity value. Other spurious effects have been shown to be produced by the pre-2003 VLBI procedures, such as applying the fixed "frame bias'' corrections to the moving pole as if they were nutation components. Although the P03 solution for the equator was shown in Capitaine et al. (2004) to be dynamically consistent and based upon both the most accurate model for the ecliptic precession and the best available models for effects of the non-rigid Earth, it was pointed out that (i) there were large uncertainties in some of the models, especially for the J₂ rate (i.e. the time derivative of the second degree zonal coefficient of the geopotential J₂) and (ii) VLBI records were at that stage unable to discriminate between the different solutions. It was therefore concluded that improvements were still needed both in the models and the fit to observations (for the integration constants) before a new model for the precession of the equator could be adopted that would represent a significant practical improvement with respect to IAU 2000.

Adopting the improved P03 ecliptic with the existing precession for the equator has been discussed as one of the possible options for replacing the current IAU precession. However, (i) retaining the IAU 2000A equator would be dynamically inconsistent due to the small dependence of the equator precession on a model for the ecliptic and (ii) new expressions for all of the angles would in any case be required. The reasons for the latter are (a) the change in the obliquity of the ecliptic at epoch, $\epsilon_0$, affects the expressions for the basic quantities for the precession of the equator, ${\omega _A}$ and ${\psi _A}$, referred to the ecliptic of epoch and (b) the direct effect of the change in the ecliptic precession on the expressions for  $\epsilon_A$ and $\chi_A$, referred to the ecliptic of date (see Table 7 of Capitaine et al. 2004) and for p_A and GST as well, which are derived from the former expressions. Therefore, the option that was retained was to adopt both an improved ecliptic precession and the best dynamically consistent solution for the precession of the equator, albeit without any noticeable improvement in predictive power so far as the CIP is concerned. During subsequent numerical consistency tests (see Appendix B of the quoted paper) it emerged that the corrections for the effects of the pre-2003 VLBI procedure should in fact be applied to the MHB values with the opposite sign than that proposed in Paper P03. Furthermore, the MHB theory had by then been slightly revised (Mathews 2004). These factors suggested that a revision of the P03 solution for the equator would be worthwhile. This view was supported by the IAU Division I Working Group on "precession and the ecliptic'' that recommended (April 2004) that the P03 solution be selected to be proposed as the new IAU precession model once its component for the precession of the equator had been revised.

This paper reports our attempts to refine the P03 solution for the equator and provides expressions for an improved model. However, the existing (unimproved) P03 solution, from Capitaine et al. (2003b, 2004), was found to have differences with respect to this improved model smaller than realistic uncertainties in the model and to fit the existing VLBI series distinctly better than the MHB 2000 or IAU 2000 models. The agreement with the improved model as well as with the VLBI results is evidence that the P03 model is trustworthy in itself. Moreover, further significant gains in accuracy would require revision of the IAU 2000A nutation model.

In view of these comparisons and of the fact that several groups already have experience in using P03 experimentally we propose that the P03 solution for both the ecliptic and equator be adopted as the replacement for the IAU 2000 precession. It is recalled in Tables 3-5.

2 Improving the P03 solution for the precession of the equator

2.1 The different steps for improving the solution

The improvement of the P03 solution for the precession of the equator was carried out in three steps, based on the set of procedures developed in the P03 paper for generating precession models but using improved observational inputs along with added refinements in the model.

As the first step, we updated the integration constants of the precession equations by using a revised MHB model (Mathews 2004) for the effects of the non-rigid Earth on precession and revised corrections for perturbing effects on the observed quantities (Capitaine et al. 2004). We then solved the P03 precession equations based on these revised integration constants and model and on the P03 solution for the ecliptic precession. This provided revised theoretical expressions for ${\psi _A}$ and ${\omega _A}$ which, except for the linear terms, differ from the P03 solution by at most a few tens of microarcseconds.

At this stage, comparisons between the precession models and VLBI data showed clearly that: (i) the addition of four years of accurate VLBI data to the 20 years available at the time of the MHB work significantly modifies the linear fit of the models to VLBI, and (ii) the difference of about -7 mas/cy² in the quadratic term of the P03 expression in ${\psi _A}$ with respect to IAU 2000 due to the J₂ rate effect considered in the P03 model has a non-negligible contribution on the estimated precession rate in longitude. We concluded that because of the strong dependence of the linear fit of the theoretical expressions to VLBI on both the span of the observations and the adopted theoretical t² term, the MHB precession rates, even when corrected for the improvement in the model and for the perturbing effects on the observed quantities mentioned above, may not be appropriate for providing the most reliable precession solution.

Therefore, as a second step, the final precession rates corresponding to the revised P03 solution were fitted to VLBI data, using the longest available interval. The quantities fitted were the residuals in the X and Y CIP coordinates, which correspond to what VLBI actually determines (as opposed to  $\Delta\psi$ and  $\Delta\epsilon$). This provided the final values for the precession rates for X and Y and therefore for ${\psi _A}$ and ${\omega _A}$.

In a third step, the finished solution was obtained by a final integration of the precession equations based on the rates found by fitting. As changes of 1 mas in the precession rates produce changes smaller than 5 microarcseconds in the coefficients of the higher-degree terms of these expressions, this last step changed the solution by no more than a few microarcseconds (apart from in the linear terms).

2.2 Upgrading the model

  For solving the precession equations (i.e. Eqs. (24) and (26) of the P03 paper), we used the revised values for the MHB "non-linear'' contributions to the precession rates (Mathews et al. 2002), recently computed by P. M. Mathews and made available to us for this purpose (Mathews 2004). These effects, denoted here by the subscript "nl'', are, in longitude and obliquity, respectively,

 

$${\rm d}r_{\psi_{\rm nl}}=-960\ \mu \rm as/cy ;\ \ \ \ d{\it r}_{\epsilon_{nl}}=+340\ \mu \rm as/cy$$ (1)

replacing the corresponding MHB values for these contributions of  $- 21~050~\mu$as/cy and $0~\mu$as/cy, respectively (cf. Table 3 of the P03 paper). This modified the quadratic term (by 10 microarcseconds) in the solution for ${\psi _A}$ and ${\omega _A}$and changed the MHB value for the dynamical flattening of the Earth and consequently the MHB correction to the precession rate in longitude, which, according to P. M. Mathews has become:

 

$$(\psi_1)_{\rm MHBrev}= (- 299~ 110 \pm 710)\ \rm\mu as/cy$$ (2)

replacing the MHB value of $(- 299~ 650 \pm 400)\ \mu$as/cy.

2.3 Upgrading the integration constants of the theoretical solution

In paper P03 (Capitaine et al. 2003b), we evaluated the corrections to be applied to the estimated MHB precession rates for some perturbing effects on the observed quantities (see expressions (32) and (33) of the quoted paper). As mentioned in the introduction, subsequent to the P03 work being carried out a re-examination of the VLBI procedures made us change our interpretation of the spurious effects of the non-rigorous pre-2003 VLBI procedure. This meant that the corrections for these effects (expression (33) of the quoted paper) had in fact to be applied to the estimated MHB values with the opposite sign than that proposed in the P03 paper (see Appendix B of Capitaine et al. 2004). The correct relationship between the integration constants in longitude and obliquity, r₀ and u₀, and the MHB estimates is (in $\mu$as/cy):

 

$$(r_0)_{\rm MHB} + 2366 - 384 = + 1982$$                            r₀ =

$$(u_0)_{\rm MHB} + 514 .$$ u₀ = (3)

The values for the precession rates derived from (i) the IAU 1976 precession rates, (ii) the revised MHB correction to these precession rates (cf. Sect. 2.2) and (iii) the revised P03 corrections to the MHB estimates, rounded off to a 5 $\mu$as level, are, in arcseconds per century:

 

                                             r₀ = 5038.7784 - 0.299110 + 0.001980 =5038.481270

u₀ = - 0.02524+ 0.000515 =-0.024725. (4)

   
Revised theoretical expressions for ${\psi _A}$ and ${\omega _A}$

The revised expressions obtained by solving the P03 precession equations with the updated values (4) for the precession rates and the changes (1) in the model for the theoretical non-linear contributions to these precession rates are:

 

$$\psi_A 5038\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...- 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{00114038}~t^3$$ =

$$+\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{0000000951}~t^5$$

$$\omega_A \epsilon_0 - 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hs... ...- 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{00772501}~t^3$$ =

$$-\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{0000003337}~t^5$$ (5)

with $\epsilon_0 = 84381\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{406}$, and:

 

$$-\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ... - 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{4297558}~t^2$$                                     X =

$$- 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}... ... 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{000007575}~t^4$$

$$+\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{0000059285}~t^5$$

$$-\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...- 22\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{4072727}~t^2$$ Y =

$$+ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}... ... 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{001112525}~t^4$$

$$+ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{0000001358}~t^5.$$ (6)

This solution will be denoted P03 $_{\rm rev1}$.

Note that the unit of time used in all the expressions is Julian century after J2000 (i.e. 1 century of TDB, or TT in practice), denoted cy.

The differences larger than 1 $\mu$as of P03 $_{\rm rev1}$ with respect to the P03 solution are, in longitude and obliquity, respectively (in $\mu$as):

$${\rm d}\psi_A = - 237~ t + 10~t^2 ;\ \ \ \ {\rm d}\omega_A = + 1029~ t.$$ (7)

The above expressions show that, except from the linear terms, the revised solution differs from the P03 solution by no more than a few tens of microarcseconds. Note that, as compared with realistic uncertainties in the precession rates (i.e. of the order of 500 $\mu$as/cy in longitude and 200 $\mu$as/cy in obliquity), only the difference in the linear term in obliquity can be considered as being significant.

   
2.5 Checks against VLBI data

Checks of four different precession models, namely MHB 2000, IAU 2000A, P03 of Capitaine et al. (2003) and P03 $_{\rm rev1}$ (i.e. expression (5) of this paper) against several series of VLBI data were carried out. The checks were based on the X and Y expressions of the components of the CIP unit vector in the Geocentric Celestial Reference System (GCRS), these being the parameters to which VLBI is directly sensitive. The coordinates X,Y were computed as the (1, 3) and (2, 3) elements, respectively, of the nutation$~\times~$precession$~\times~$bias matrix, using the precession model in question and the IAU 2000A nutation and frame bias. The "MHB 2000'' model corresponds to a straightforward use (similar to the original MHB model) of the MHB 2000 precession rates and frame biases as components of the MHB 2000A nutation. The other models correspond to a rigorous use of the MHB 2000 frame biases, precession expressions and MHB 2000A nutation in the individual rotations for biases, precession and nutation, respectively.

The observed X,Y were derived in the same way, namely by using the IAU 1976 precession and adding the VLBI celestial pole offsets to the IAU 1980 nutation. We verified that the fits of the models against observations were identical when based on either (a) the residuals in the X and Y parameters obtained as described above or (b) the residuals of the differences between the models for longitude $\times \sin\epsilon_0$ and obliquity with respect to IAU 1976/1980 and the series of VLBI observed "celestial pole offsets''.

We used in these fits individual solutions from the Goddard Space Flight Center (GSFC), US Naval Observatory (USNO) and Institute of Applied Astronomy (IAA) and also the C04 combined solution of the IERS Product Center. These series were used for various intervals from 1980.0 to 2004.3.

Preliminary checks revealed that:

-

the IAU 2000A precession-nutation model, not surprisingly, gives a much better fit to the data than IAU 1976/1980;

-

there is no significant difference in the overall rms corresponding to the fit of the above four models against VLBI observations;

-

fitting straight lines to the residuals between the models and the observations give results that are significantly different according to which precession model is used;

-

the time span of the data is too short to allow a reliable quadratic fit in the residuals between the models and the observations;

-

changing the quadratic term gives a significant difference in the estimated linear fit because the data interval is asymmetric about t = 0;

-

the VLBI data before 1987 are excessively noisy;

-

the free core nutation (FCN) must be corrected to avoid systematic errors in the linear fits;

-

the C04 solution shows large structures and noise that do not appear in the individual series.

We concluded that:

-

the limited span of the VLBI data prevents us estimating the corrections to both the precession rate and the quadratic term in longitude;

-

as the quadratic term in longitude is strongly dependent on the J₂ rate model, the linear fits are dependent on the value chosen for this parameter and this effect has to be taken into account;

-

even if VLBI cannot at present allow us to discriminate between these precession models, linear fits of the models against observations might yield interesting information;

-

when performing the fits, it is essential to weight the observations according to their formal errors, in order to benefit from both the longest time spans and the most precise and recent observations;

-

the inhomogeneous properties of the C04 IERS solution for the celestial pole offsets, which is a combination of several individual VLBI series of these parameters, makes this series less appropriate for such fits than the individual series.

The first step was to remove from the observations the FCN, which appears as a term of about 430 days period with a varying amplitude and phase. We fitted sine and cosine terms at the FCN period to data series of (in most cases) two years, verified that the results agreed with those supplied in the MHB_2000 Fortran code up to 2000 and then extended the model for a further four years with our own results (see Table 2). We used the extended model to remove from the observations the FCN effects, prior to performing a linear least-squares fit. Each observation, i, was weighted according to its formal error provided in the VLBI series, $(\sigma_{\psi})_i$, $(\sigma_{\omega})_i$ in longitude and obliquity, respectively. The weights in X and Y were computed as: ( $W_{X})_i=1/(\sigma_{\psi} \sin\epsilon_0)_i^2$, ( $W_{Y})_i=1/(\sigma_{\omega})_i^2$.

We selected two different spans of data: (i) the interval 1980.0-2000.0, which was the interval used for the original MHB fit, and (ii) the interval 1980.0-2004.3, which is the longest available and has an additional four years of low-noise data. We also made some complementary linear fits for the interval 1990.0-2004.3. Results for these various fits are set out in Table 1.

Figures 1 to 10 plot the differences between the MHB 2000, ${\rm P}03_{\rm rev1}$ or P03 solutions and the GSFC, IAA or C04 VLBI series, over the interval 1980.0 to 2004.3 (Figs. 1 to 5 for X (i.e.  $\Delta\psi \sin\epsilon_0$) and Figs. 6 to 10 for Y (i.e.  $\Delta\omega$)).

Table 1: Linear fits (d $X_0 + {\rm d}X_1 t;\ {\rm d}Y_0 + {\rm d}Y_1 t$) of precession models for the $X, \ Y$ coordinates of the CIP in the GCRS against VLBI data; unit: microarcsecond.

These figures and Table 1 show the following results.

-

Fits of MHB 2000 to the GSFC and USNO VLBI series that were used in the original MHB fits (i.e. identified as GSF1122 and usn9901 IVS/IERS series according to Herring et al. (2002) and denoted here GSFC $_{\rm MHB}$ and USNO $_{\rm MHB}$), are in very good agreement with the MHB results, giving constant differences less than 50 $\mu$as and linear differences less than 110 $\mu$as/cy in both X and Y.

-

There is a similar agreement of the fits of MHB 2000 to revised GSFC and USNO series for the same interval and good agreement with all the VLBI series for the period 1980-2000: a linear fit to the residuals of any of these series provides a slope smaller than 100 $\mu$as/cy for X and 400 $\mu$as/cy for Y.

-

There are significant differences between results for the intervals 1990.0-2004.3 and 1980.0-2004.3 with respect to those for the same models and same VLBI series, but for the interval 1980.0-2000.0.

-

GSFC and USNO series provide results that are equivalent within the formal errors of the fits; therefore retaining one of them will be sufficient for the present work.

-

The differences between the fits of IAU 2000 and MHB 2000 correspond, as expected, to the difference between the pre-2003 and post-2003 VLBI procedures (i.e. +152 $\mu$as/cy for X and -514 $\mu$as/cy for Y).

-

The differences between the fits of P03 and IAU 2000 include, as expected, (i) the correction applied in P03 for correcting the effect of the pre-2003 VLBI procedure, (ii) the effect of the difference in the quadratic term in X between P03 and IAU 2000.

-

The P03 $_{\rm rev1}$ solution in obliquity gives exactly the same results as MHB 2000, which proves that this solution is reproducing a precession rate that, when based on an implementation that is different from MHB (due to the change in obliquity at epoch, frame bias handling, etc.), reproduces the MHB model.

-

The P03 $_{\rm rev1}$ solution in longitude does not give similar results to MHB 2000, as might be expected given that (i) it is based on the MHB $_{\rm rev}$ precession rate which differs by 540 $\mu$as/cy from the MHB value and (ii) there is the additional effect due to the difference in the t² term between the two solutions.

-

The linear fit of MHB 2000 with respect to VLBI over this interval 1980.0-2004.3 is of the order of 700 $\mu$as/cy (for X) and 900 $\mu$as/cy for Y, using several series of VLBI data. Note that such large residual linear terms in the VLBI fit of MHB 2000 over the longest available span of data is consistent with comparisons reported by Malkin (2004).

-

The P03 solution fits the VLBI series distinctly better than the MHB 2000, IAU 2000 and P03 $_{\rm rev1}$ models.

We also performed similar fits for the Bretagnon et al. (2003) model and saw the expected discrepancies with respect to the fits of the IAU 2000 model: i.e. the use of a value for $\epsilon_0$ which is not appropriate for the MHB precession rate in longitude has produced a -0.9 mas t effect in X while the use of a rigid Earth model has led to a -1.3 mas t effect in Y. The additional discrepancy with respect to the fits of the P03 model comes from the fact that this model contains a -7 mas/cy² contribution for the J2 rate.

	Figure 1: X differences, MHB 2000 minus GSFC.
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	Figure 2: X differences, MHB 2000 minus IAA.
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	Figure 3: X differences, P03 $_{\rm rev1}$ minus GSFC.
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	Figure 4: X differences, P03 minus IAA.
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	Figure 5: X differences, P03 $_{\rm rev1}$ minus C04.
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	Figure 6: Y differences, MHB 2000 minus IAA.
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	Figure 7: Y differences, P03 $_{\rm rev1}$ minus GSFC.
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	Figure 8: Y differences, P03 $_{\rm rev1}$ minus C04.
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	Figure 9: Y differences, P03 minus IAA.
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	Figure 10: Y differences, P03 minus GSFC.
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2.6 Revised expressions based on precession rates fitted to VLBI

We have concluded from the various VLBI checks that the MHB values do not fit well with the most recent and accurate VLBI data. Therefore it appeared that it might be preferable that the final precession solution rely on more recent VLBI fits (1980.0-2004.3) than applying theoretical corrections to the MHB precession rates (that result from a fit to VLBI data on the interval 1980-2000). For the final estimates of the correction to the precession rates of the models, we have retained only the GSFC and IAA series which (i) both cover the longest time span and (ii) are based upon independent reduction software (viz. CALC for GSFC and OCCAM for IAA).

Table 2: Amplitudes of the FCN mode fitted to VLBI data; unit: milliarcsecond. For the fits made in this paper, the MHB model was used up to and including 2000/01/01, and the fitted coefficients after this date.

The estimated correction to the P03 $_{\rm rev1}$ solution is:

$${\rm d}X = + 326\ \mu{\rm as/cy} ;\ \ \ \ {\rm d}Y = - 950\ \mu\rm {as/cy}$$ (8)

from which we can derive the correction to the precession rates in ${\psi _A}$ and ${\omega _A}$ corresponding to the IERS 2003 value for d$\alpha_0$of -14.6 mas:

$${\rm d}\psi_A= + 820\ \mu{\rm as/cy} ;\ \ \ \ {\rm d}\omega_A = {\rm d}\epsilon_A= - 950\ \mu\rm {as/cy}$$ (9)

and therefore:

 

$$\psi_1 5038\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...pace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{025675} /{\rm cy} ,$$ =

$$\epsilon_1 -46\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{836690} /{\rm cy}.$$ = (10)

We developed a solution like P03 $_{\rm rev1}$ in Sect. 2.2 by solving again the P03 precession equations but this time using as integration constants the precession rates in longitude and obliquity derived from VLBI fits given by Eq. (10). The final expressions, that we will denote "the P03 $_{\rm rev2}$ solutions'', are:

 

$$\psi_A 5038\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...- 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{00114040}~t^3$$ =

$$+\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{0000000951}~t^5$$

$$\omega_A \epsilon_0 - 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hs... ...- 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{00772501}~t^3$$ =

$$-\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{0000003337}~t^5$$ (11)

with $\epsilon_0 = 84381\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{406}$, and:

 

$$-\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ... - 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{4297752}~t^2$$                                   X =

$$- 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}... ... 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{000007576}~t^4$$

$$+\ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{0000059285}~t^5$$

$$- 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}... ...- 22\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{4072801}~t^2$$ Y =

$$+ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}... ... 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{001112526}~t^4$$

$$+ 0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{0000001358}~t^5.$$ (12)

The linear fits of the P03 $_{\rm rev2}\ X, Y$ expressions to VLBI over the time range of 1980.0-2004.3 are:

(i)

using the IAA series:
${\rm d}X=(-45 \pm 3) +\ (+ 21 \pm 60)\ t\\ {\rm d}Y=(+39 \pm 3)+\ (-137 \pm 60)\ t;$

(ii)

using the GSFC series:
${\rm d}X=(-8 \pm 3) +\ (+ 72 \pm 55)\ t\\ {\rm d}Y=(+73 \pm 3)+\ (+ 124 \pm 60)\ t;$

and the differences larger than 1 $\mu$as of P03 $_{\rm rev2}$ with respect to the P03 solution are, in $\mu$as:

$${\rm d}\psi_A = + 583~ t + 15~t^2 ;\ \ \ \ {\rm d}\omega_A = + 79~ t ,$$ (13)

or equivalently:

$${\rm d}X= + 232~ t + 8~t^2 ;\ \ \ \ \ \ \ \ {\rm d}Y= + 79~ t - 5~t^2 .$$ (14)

We notice that this solution (i) fits well to the longest VLBI series and (ii) is very close to the P03 solution both in X and Y.

   
2.7 Precision of the coefficients

Regarding the precession of the ecliptic, we recall from Sect. 1 that the accuracy of the secular variations of the P03 solution in P_A and Q_A was estimated to be about 0.05 mas/cy over a two-millennium interval centered on J2000.0.

Table 3: The P03 primary precession quantities compared with the IAU 2000 solution (IAU); unit: milliarcsecond.

Regarding the precession of the equator we have already mentioned that, although the precision in computing the precession solution has been significantly improved compared with that of the IAU 2000 precession by (i) using the best available model for the ecliptic precession, (ii) using the best available model for the non-rigid Earth effects and (iii) providing a solution that is dynamically consistent, this model is dependent on the model for some parameters of the non-rigid Earth, such as the J₂ rate, that have large uncertainties. The accuracy of the expression for the precession in longitude is therefore strongly limited by the uncertainty in the model for the J₂ time variations, which is if the order of 20% (Bourda & Capitaine 2004), resulting in uncertainties of the order of 1.5 mas/cy² in the quadratic term in ${\psi _A}$. As noted in that paper, the J₂ rate value used for the P03 solution (i.e. of the order of $-3 \times 10^{-9}$/cy) is in good agreement with the J₂ rate evaluation of  $\left( -3.4\pm0.6 \right) \times 10^{-9}$/cy by Morrison & Stephenson (1997) from long term studies of the Earth rotation variations, based upon eclipse data over two millennia.

Moreover, the linear fit to VLBI is still spanning too small an interval to provide a realistic uncertainty better than about 150 $\mu$as/cy for X and Y despite the smaller formal uncertainty provided by the fit.

2.8 Choice of the precession solution for the equator

There are three choices for replacing the IAU 2000 precession of the equator, all dynamically consistent: (i) use P03 $_{\rm rev1}$, based on the MHB precession rates but corrected for various theoretical effects; (ii) use P03 $_{\rm rev2}$, based on the latest VLBI estimated precession rates; and (iii) retain the existing P03 solution (the small change in the t² term from P03 to P03 $_{\rm rev2}$ being in fact smaller than the uncertainty in this term). Given the results of Sects. 2.5 and 2.6, and the evaluation of the precision provided in Sect. 2.7, choice (iii) (i.e. retain P03) seems the most appropriate as long as it is associated with a parameterized solution as a function of the parameters (i) to which the precession expressions are the most sensitive and (ii) which are expected to be improved in the future.

3 Final expressions for the new precession model

The final expressions proposed here as replacements for the IAU 2000 precession quantities include (i) P03 expressions for the ecliptic precession; (ii) P03 expressions for the primary precession quantities for the equator; (iii) precession quantities for classical use derived from the previous P03 quantities including revised sidereal time and (iv) expressions for precession quantities for use with the new paradigm based on the positions of the celestial intermediate pole (CIP) and celestial intermediate origin (CIO), respectively, in the GCRS. These P03 numerical expressions are associated with a parameterized extension comprising functions of the J2 rate model and of the changes to the P03 precession rates.

Note that the coefficients of the P03 expressions are given in the tables with more digits than the real uncertainty, for the purpose of internal consistency.

Table 4: The P03 expressions for the derived precession quantities for classical use and Greenwich mean sidereal time expressed in terms of the Earth rotation angle $\theta$; unit: milliarcsecond.

Table 5: The P03 expressions for the GCRS components of the CIP and angular position of the CIO, and for the rotation vector components; unit: milliarcsecond.

3.1 P03 expressions for the primary precession quantities

The most "fundamental'' parameters for the precession of the equator are those (either angular variables or Cartesian coordinates) providing the "dynamically derived'' position of the equator (or equivalently the CIP) relative to the fixed ecliptic. These are indeed the most basic variables due to the fact that the most recent analytical theories for the solar system bodies, upon which the Earth's rotation theory is based, are referred to the fixed ecliptic. The dynamics of Earth rotation do not involve the concept of an ecliptic, but are directly based on the dynamics of the solar system bodies in an inertial frame, which means that using the ecliptic as a reference is only an approximate intermediate stage in obtaining simplified expressions for the precession-nutation of the equator. Suitable precession parameters for the ecliptic are the (x, y) coordinates of the ecliptic pole in the mean ecliptic frame of epoch. Note that the x, y coordinates (either of the ecliptic pole or the equatorial pole) are more basic quantities than individual angles that are determined by their trigonometric relations.

The primary precession quantities selected for the P03 solution, from which all the other quantities can be derived, are:

(i)

the expressions for the ecliptic quantities P_A, Q_A which may be regarded as, respectively, the x and -y components of the secularly-moving ecliptic pole vector in a (right-handed) coordinate system that has its x-axis through the J2000 (inertial) mean equinox and its z-axis through the J2000 ecliptic pole;

(ii)

the expressions for the quantities ${\psi _A}$, ${\omega _A}$ for the equator, which are the polar coordinates of the CIP with respect to the J2000 ecliptic pole and J2000 (inertial) mean equinox.

The developments for these primary quantities are provided in Table 3 together with the corresponding IAU 2000 expressions.

3.2 P03 expressions for the derived precession quantities for classical use and GMST

The P03 expressions for the ecliptic precession angles $\pi_A$ and $\Pi_A$ were derived from the developments for the basic ecliptic quantities P_A and Q_A, which can be written as $P_A=\sin\pi_A\sin\Pi_A$ and  $Q_A=\sin\pi_A\cos\Pi_A$. Similarly, the P03 expressions for the two first equatorial precession angles $\zeta_A$ and $\theta_A$ were derived from the P03 developments for ${\psi _A}$ and ${\omega _A}$.

The expressions for all the precession parameters that refer to the ecliptic of date (i.e. the third equatorial precession angle, z_A, the obliquity of the equator on the moving ecliptic, $\epsilon_A$, the planetary precession along the equator of date, $\chi_A$ and the general precession in longitude, p_A) were derived from both the P03 ecliptic and equator primary expressions.

Table 4 provides expressions for all the precession quantities for classical use, along with the expression for GMST(UT1,TT) the IAU 2000 expression for which (Capitaine et al. 2003a) must be revised in order to take account of the changes in the expressions for the precession quantities ${\psi _A}$, $\chi_A$ and ${\omega _A}$. Note that, in contrast, the expressions for the complementary terms in the equation of the equinoxes are unchanged.

3.3 P03 expressions for precession quantities for use with the new paradigm

A replacement for the IAU model must provide improved precession in both the classical and new paradigms. The new paradigm provides the CIP directly, without any concept of "mean pole''. If "precession'' is to be a meaningful and useful concept in the new paradigm, we should consider precession of the equator as being the secular part of the "orientation parameters'' of the CIP equator with respect to either J2000 mean equatorial system or the GCRS. The precession expressions for X and Y are thus identified with the polynomial part of the GCRS CIP direction cosines, based on the P03 expressions for ${\psi _A}$ and ${\omega _A}$.

The P03 expressions for these quantities as well as for the quantity s+XY/2 that provides the GCRS position of the CIO are given in Table 5. Note that the only significant change with respect to the IAU 2000 expression for the GCRS position of the CIO is of $2.7~\mu$as in the quadratic term, the other changes all being less than $0.5~\mu$as. Table 5 also provides the components $x_{\rm r}$, $y_{\rm r}$, $z_{\rm r}$ of the "rotation vector'' representing bias plus the P03 precession (see Capitaine et al. 2003b). The rotation vector is a concise and direct way to represent the rotation of the coordinate system, in this case from GCRS axes to mean equator and equinox of date. Its direction is the pole of the rotation (the Euler axis) and its magnitude is the amount of rotation (the Euler angle). In this case, to first order, the rotation vector points at the ecliptic pole and its magnitude increases steadily by about 50 arcsec per year.

3.4 The parameterized P04 precession expressions

Due to the strong dependence of (i) the precession expressions on the precession rates r₀ and u₀ in longitude and obliquity, respectively and (ii) the precession in longitude (or equivalently in the GCRS CIP X coordinate) on the J₂ rate model, both of which are expected to be improved in the future, we have developed a parameterized solution for these quantities as a function of those parameters, calling it P04 $_{\rm par}$. Such a solution is intended to be used to produce (or check) future precession models based on extended VLBI records and improved geophysical models. It is based on the P03 precession expressions (Capitaine et al. 2003b) and on the theoretical relationship of their linear, quadratic, and cubic coefficients, $\psi_i$, $\omega_i$, $\epsilon_i$, $\chi_i$ (with i=1, 2, 3 for the linear, quadratic and cubic coefficients, respectively), with constant and linear contributions, r₀, u₀, r₁, u₁ to the precession rates, $r_{\psi}$ in longitude and $r_{\epsilon}$ in obliquity. The theoretical relationships were derived from Tables 3 and 7 of the P03 Paper using the following expressions:

$$\psi_1 r_0 ;\ \ \ \omega_1=u_0$$ =

$$\psi_2 \frac{1}{2} \left(r_1 +r_0c_1\cot\epsilon_0 -\frac{u_0s_1}{\sin^2\epsilon_0}\right)$$ =

$$\omega_2 \frac{1}{2}(u_1+r_0s_1)$$ = (15)

and:

$$\chi_1 s_1/\sin\epsilon_0 ;\ \ \ \epsilon_1=c_1+u_0$$ =

$$\chi_2 \frac{1}{\sin\epsilon_0} (s_2 +r_0c_1-s_1 \cot\epsilon_0 (u_0+c_1))$$ =

$$\epsilon_2 c_2+ \frac{1}{2}(u_1-r_0s_1+s_1^2\cot\epsilon_0)$$ = (16)

s₁, c₁ and s₂, c₂ being the linear and quadratic coefficients of the precession ecliptic quantities P_A and Q_A, respectively, and:

$$r_{01}+r_{02}=f_{01}\cos\epsilon_0 + r_{02}$$                                 r₀ =

$$u_{01}+u_{02}=g_{01}\cos\epsilon_0 + u_{02},$$ u₀ = (17)

$$-f_{01} (u_0 + c_1)\sin\epsilon_0 + f_{11}\cos\epsilon_0 + r_{12}$$                               r₁ =

$$-g_{01} (u_0 + c_1)\sin\epsilon_0 + g_{11}\cos\epsilon_0 + u_{12}.$$ u₁ = (18)

$(f_{01} + f_{11}~t )\cos\epsilon_0$ and $(g_{01} + g_{11}~t) \cos\epsilon_0$ being the first-order luni-solar contributions to $r_{\psi}$ and $r_{\epsilon}$, respectively. Note that these components are such that f₀₁ and g₀₁ are proportional to J₂ and f₁₁ and g₁₁ contain a part proportional to $\dot{J_2}$.

This gives for ${\psi _A}$ and ${\omega _A}$:

$$\psi_2 \frac{1}{2}\Big[-f_{01} u_0 \sin\epsilon_0 + f_{11}\cos\epsilon_0 + r_{12} -\frac{u_0s_1}{\sin^2\epsilon_0}$$ =

$$+f_{01} c_1(\cos^2\epsilon_0-\sin^2\epsilon_0)/\sin\epsilon_0 + r_{02}c_1 \cot\epsilon_0\Big]$$

$$\omega_2 \frac{1}{2}\big[-g_{01} (u_0 + c_1)\sin\epsilon_0 + r_0s_1$$ =

$$+ g_{11}\cos\epsilon_0 + u_{12}\big],$$ (19)

and similar relationships for the other precession quantities.

The parameterized expressions of the precession quantities as functions of $\dot{J_2}$ and of corrections dr₀, du₀ to the P03 precession rates r₀ and u₀, respectively, should retain only the parameterized terms that, given the expected values for the parameters considered, can contribute to the expressions with amplitudes larger than one microarcsecond. The time derivative $\dot{J_2}$ of the coefficient J₂ contributes to r₁ as follows:

$$r_1(J2d)= f_{11}(J2d) \cos\epsilon_0 = (\dot{J_2}/J_2)f_{01}\cos\epsilon_0.$$ (20)

According to Table 7 of the P03 Paper and to expected values lower than 1 mas/cy for the corrections to the precession rates and $5 \times 10^{-6}$ for the term  $\dot{J_2}/J_2$(see Sect. 3.6), this requires considering the following terms in the partials of the coefficients of the precession expressions:

$$\partial{\psi_1} /\partial{r_0} 1 ;\ \partial{\omega_1} /\partial{u_0}=\partial{\epsilon_1} /\partial{u_0}=1,$$ =

$$\partial{\psi_2} /\partial{u_0} \approx -\frac{1}{2}r_{01} \tan\epsilon_0;\ \partial{\psi_2} /\partial{r_1}=\frac{1}{2},$$

$$\partial{\psi_3} /\partial{r_1} \frac{1}{3}c_1\cot\epsilon_0; \partial{\chi_2} /\partial{r_0}=2\partial{\chi_3} /\partial{r_1}=\frac{c_1}{\sin\epsilon_0}\cdot$$ = (21)

The resulting P04 $_{\rm par}$ expressions for classical use are, with dr₀, du₀ in arcseconds:

 

$$\psi_A (P04_{\rm par}) \psi_A (P03)+ {\rm d}r_0~t -0.0053~ {\rm d}u_0~t^2$$ =

$$+ \ \left[0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspac... ...space{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{000002}~t^3\right]$$

$$+ (\dot{J_2}/J_2)~ \times \left(2520\hspace{0.1em}'\hspace{-0.06e... ... -0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{9}~t^3\right)$$

$$\omega_A(P04_{\rm par}) \omega_A(P03) + {\rm d}u_0~t ,$$ = (22)

$$\chi_A(P04_{\rm par}) \chi_A(P03) - 0.0006~ {\rm d}r_0~t^2$$ =

$$- [0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... .../J_2)~ (1\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{4}~t^3)$$

$$\epsilon_A(P04_{\rm par}) \epsilon_A(P03) + {\rm d}u_0~t .$$ = (23)

And the expression for Greenwich mean sidereal time is:

$$GMST(P04_{\rm par})$$ = GMST(P03)

$$+ {\rm d}r_0 \cos\epsilon_0~t -0.0098~{\rm d}u_0~t^2$$

$$+ \ \left[0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspac... ...space{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{000002}~t^3\right]$$

$$+ (\dot{J_2}/J_2)~ \times \left(2312\hspace{0.1em}'\hspace{-0.06e... ...+0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{6}~t^3\right).$$ (24)

The additional t² and t³ terms that appear between square brackets in the expressions for the differences (P04 $_{\rm par}-$P03) for the precession quantities ${\psi _A}$, $\chi_A$ and $\it GMST$ (and also X, Y below) are for removing the $\dot{J_2}/J_2$ contribution to the P03 solution. Taking into account additionally the effects of the corrections d$\xi_0$, d$\eta_0$and d(d$\alpha_0)$ to the IAU 2000 frame biases, we get the parameterized P04 expressions for the GCRS X and Y quantities for use with the new paradigm:

 

$$X(P04_{\rm par}) X(P03) + {\rm d}\xi_0 + 0.0001~{\rm d}({\rm d}\alpha_0)~t^2$$ =

$$+ {\rm d}X_1~t +~0.0203~ {\rm d}u_0~t^2$$

$$+ \left[0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{... ...space{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{000001}~t^3\right]$$

$$+ (\dot{J_2}/J_2)~ \times (1002\hspace{0.1em}'\hspace{-0.06em}'\h... ...5}~t^2 -0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{4}~t^3)$$

$$Y(P04_{\rm par}) Y(P03) + {\rm d}\eta_0 + X_1\ {\rm d}({\rm d}\alpha_0)~t$$ =

$$+ {\rm d}u_0~t -0.0224~ {\rm d}X_1~t^2$$

$$- [0\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em... ...times (22\hspace{0.1em}'\hspace{-0.06em}'\hspace{-0.3em}.\hspace{0.1em}{5}~t^3)$$ (25)

where dX₁ (= ${\rm d}r_0 \sin\epsilon_0$) is the correction to the linear term X₁ of the P03 X expression.

3.5 Discussion on the choice of the basic parameters for precession-nutation

Suitable precession-nutation parameters would integrate the computation of bias, precession and nutation and provide a transformation between celestial and terrestrial coordinates that involves a minimum number of variables and coefficients.

Suitable candidates for replacing the classical precession angles ${\psi _A}$, ${\omega _A}$, $\chi_A$ and nutation angles $\epsilon_A$, $\Delta\psi$, $\Delta\epsilon$, which are usually considered separately, are the combined precession-nutation angles (Aoki & Kinoshita 1983) $(\psi_A+ \Delta\psi_1)$, $(\omega_A+\Delta\epsilon_1)$ referred to the ecliptic of epoch, which have the advantages:

(i)

of being the "fundamental parameters'' mentioned above for precession alone, associated with the parameter  $(\chi_A+ \Delta\chi_A)$ for positioning the true equinox along the equator; note that the last quantity is not in fact necessary if it is omitted both in the PN matrix and in the expression for GST; note also that the combination of precession and nutation could easily include the frame biases as well;

(ii)

of being referred to an inertial frame which is more in agreement with the IAU adoption of the International Celestial Reference System (ICRS) than referring to a moving ecliptic that involves taking additional precession effects into account.

The use of such parameters would mean using solutions for nutation that are directly referred to the mean ecliptic at epoch instead of ecliptic of date. Note that such an approach was already followed by Woolard (1953) and Bretagnon et al. (1997) and would be the natural choice for any numerical integration.

In the new paradigm, the basic quantities are the CIP coordinates X, Y in the GCRS, that appear directly as the (1, 3) and (2, 3) elements of the celestial-to-terrestrial transformation matrix. The polar coordinates E and d in the GCRS can easily be derived from X and Y.

A third option is to use the "rotation vector'' approach (see Sect. 3.3), once it has been extended to include nutation as well.

   
3.6 Changes in the IAU nutation corresponding to the adoption of the P03 precession

Adopting a new precession model requires slight adjustment in the amplitudes of the nutation model that is used. The effects to be considered are the following:

(i)

 a change of the amplitudes of nutation in longitude due the change in the obliquity of the ecliptic in the precession model as compared with its value when estimating the nutation amplitudes of the model, similar to the effect considered for the P03 precession as compared with the IAU 2000 precession; in order to take this change into account, it is necessary to multiply the amplitudes of the nutation in longitude by $\sin\epsilon_{{\rm IAU2000}}/\sin\epsilon_{{\rm P03}} = 1.000000470$;

(ii)

 a change of the amplitudes of nutation (both in longitude and obliquity) due to the secular variation of the Earth's dynamical flattening (or equivalently J₂) which is used in the P03 precession model, whereas it was not considered in the IAU 2000 model; the amplitudes of nutation being proportional to J₂, this J₂rate effect gives rise to additional Poisson terms that are proportional to $\dot{J_2}/J_2$. The J₂ rate contribution to the linear precession rate in longitude, which was considered in the P03 precession to be -14 mas t, corresponds to a value for $\dot{J_2}/J_2$ of  $-2.7774\times 10^{-6}$, or equivalently (with $J_2 =1.0826358 \times 10^{-3}$) a value of $-0.3001 \times 10^{-9}$/cy.

The corrections larger than 1 $\mu$as to be added to the IAU 2000 nutation for these above effects are, respectively, in $\mu$as:

$${\rm d}_1\psi=-8.1\ \sin\Omega -0.6\ \sin(2F-2D+2\Omega)$$ (26)

 

$${\rm d}_2\psi +~ 47.8~ t~ \sin\Omega +~ 3.7~ t~ \sin(2F-2D+2\Omega)$$ =

$$+ 0.6~ t~ \sin(2F+2\Omega) -0.6~ t~ \sin2 \Omega$$

$${\rm d}_2\varepsilon ~ -25.6~ t~ \cos\Omega -1.6~ t~ \cos(2F-2D+2\Omega).$$ = (27)

Expressions (27) can also be provided in arcseconds in the following parameterized form:

$${\rm d}_2\psi (\dot{J_2}/J_2)\ t~ [-17.2~ \sin\Omega - 1.3~ \sin(2F-2D+2\Omega)$$ =

$$- 0.2~ \sin(2F+2\Omega) + 0.2~ \sin2 \Omega]$$

$${\rm d}_2\varepsilon (\dot{J_2}/J_2)\ t~ [9.2~ \cos\Omega + 0.6~ \cos(2F-2D+2\Omega)].$$ = (28)

The above corrections can be written as functions of $\Delta\psi_{{\rm IAU2000}}$ and $\Delta\varepsilon_{{\rm IAU2000}}$, the IAU 2000 nutation angles in longitude and obliquity respectively:

$${\rm d}_1\psi= [(\sin\epsilon_{{\rm IAU2000}}/\sin\epsilon_{P03}) - 1]\ \Delta\psi_{{\rm IAU2000}}$$ (29)

$${\rm d}_2\psi= (\dot{J_2}/J_2)\ t~ \Delta\psi_{{\rm IAU2000}}$$

$${\rm d}_2\varepsilon = (\dot{J_2}/J_2)\ t~ \Delta\varepsilon_{{\rm IAU2000}}.$$ (30)

4 Summary

In this paper we have provided expressions for precession as possible replacements for the current IAU precession (adopted by IAU 2000 Resolution B1.6, Mathews et al. 2002).

The proposed expressions are the P03 solution of Capitaine et al. (2003b) for the ecliptic and the equator. We have in addition provided a parameterized P04 solution for the equator that could be used for future improvement. Before recommending the P03 solution for the equator, we compared it with a revised version of the solution, which is based on the P03 ecliptic and was obtained in exactly the same way as P03 but using a refined model for the contributions of the non-rigid Earth (Mathews 2004) and revised integration constants for the precession rates that came from fits to the latest and most reliable VLBI data. After considering several options for the final precession expressions, we concluded that retaining the P03 solution, which was already in experimental use, was the preferred option.

In this paper we have described the different steps in obtaining the improved solution and in making comparisons, and we have summarized the expressions for all the usual precession quantities as well as for sidereal time and for the quantities to be used in the CIO-based transformation. We have also provided the corresponding corrections to be applied to the IAU 2000 nutation.

Acknowledgements

We are grateful to P. M. Mathews for having provided his latest pre-publication results for the effects of the non-rigid Earth on precession, and for valuable discussion. We thank J. Hilton, G. Kaplan and J. Vondrák for their interest and encouragement and V. Dehant for helpful comments and suggestions for improving the manuscript.

References

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