G. Bourda - N. Capitaine
Observatoire de Paris, SYRTE/UMR 8630-CNRS, 61 avenue de l'Observatoire, 75014 Paris, France
Received 25 June 2004 / Accepted 10 August 2004
Abstract
Precession and nutation of the Earth depend on the Earth's dynamical flattening, H, which is closely related to the second degree zonal coefficient, J_{2} of the geopotential. A small secular decrease as well as seasonal variations of this coefficient have been detected by precise measurements of artificial satellites (Nerem et al. 1993; Cazenave et al. 1995) which have to be taken into account for modelling precession and nutation at a microarcsecond accuracy in order to be in agreement with the accuracy of current VLBI determinations of the Earth orientation parameters. However, the large uncertainties in the theoretical models for these J_{2} variations (for example a recent change in the observed secular trend) is one of the most important causes of why the accuracy of the precession-nutation models is limited (Williams 1994; Capitaine et al. 2003). We have investigated in this paper how the use of the variations of J_{2} observed by space geodetic techniques can influence the theoretical expressions for precession and nutation. We have used time series of J_{2} obtained by the "Groupe de Recherches en Géodésie spatiale'' (GRGS) from the precise orbit determination of several artificial satellites from 1985 to 2002 to evaluate the effect of the corresponding constant, secular and periodic parts of H and we have discussed the best way of taking the observed variations into account. We have concluded that, although a realistic estimation of the J_{2} rate must rely not only on space geodetic observations over a limited period but also on other kinds of observations, the monitoring of periodic variations in J_{2} could be used for predicting the effects on the periodic part of the precession-nutation motion.
Key words: astrometry - reference systems - ephemerides - celestial mechanics - standards
Expressions for the precession of the equator rely on values for the precession rate in longitude that have been derived from astronomical observations (i.e. observations that were based upon optical astrometry until the IAU1976 precession, and then on Very Long Baseline Interferometry (VLBI) observations for more recent models). The IAU2000 precession-nutation model provided by Mathews et al. (2002) (denoted MHB 2000 in the following), that was adopted by the IAU beginning on 1 January 2003, includes a new nutation series for a non-rigid Earth and corrections to the precession rates in longitude and obliquity that were estimated from VLBI observations during a 20-year period. The precession in longitude for the equator being a function of the Earth's dynamical flattening H, observed values for this precession quantity are classically used to derive a realistic value for H. Such a global dynamical parameter of the Earth is generally considered as a constant, except in a few recent models for precession (Williams 1994; Capitaine et al. 2003) or nutation (Souchay & Folgueira 1999; Mathews et al. 2002; Lambert & Capitaine 2004) in which either the secular or the zonal variations of this coefficient are explicitly considered through simplified models.
The recent implementation of the IAU2000 A precession-nutation model guarantees an accuracy of about 200 as in the nutation angles, and all the predictable effects that have amplitudes of the order of 10 as have therefore to be considered. One of these effects is the influence of the variations () in the Earth's dynamical flattening, which are not explicitly considered in the IAU2000 A precession-nutation model. Furthermore, the IAU2000 precession is based on an improvement of the precession rates values derived from recent VLBI measurements, but it does not improve the higher degree terms in the polynomials for the precession angles , of the equator (see Fig. 1). This precession model is not dynamically consistent because the higher degree precession terms are actually dependent on the precession rates (Capitaine et al. 2003) and need to be improved, even though VLBI observations are unable to discriminate between recent solutions due to the limited span of the available data (Capitaine et al. 2004). One alternative way for such an improvement is to improve the model for the geophysical contributions to the precession angles and especially the influence of (or equivalently ).
The H parameter is linked to the dynamical form-factor, J_{2} for the Earth (i.e. the C_{20} harmonic coefficient of the geopotential) which is determined by space geodetic techniques on a regular basis. Owing to the accuracy now reached by these techniques, the temporal variation of a few Earth gravity field coefficients, especially , can be determined (for early studies, see for example Nerem et al. 1993; Cazenave et al. 1995; or Bianco et al. 1998). They are due to Earth oceanic and solid tides, as well as mass displacements of geophysical reservoirs and post-glacial rebound for . This coefficient C_{20} can be related to the Earth's orientation parameters and more particularly to the Earth precession-nutation, through H. The purpose of this paper is to use space geodetic determination of the geopotential to estimate , in order to investigate its influence on the precession-nutation model. The C_{20} data used in this study have been obtained from the positioning of several satellites between 1985 and 2002. We estimate also the constant part of H, based on such space geodetic measurements, and compare its value and influence on precession results with respect to those based on VLBI determinations.
In Sect. 2 we recall the equations expressing the equatorial precession angles as a function of the dynamical flattening H. We provide the numerical values implemented in our model, compare the values obtained for H by various studies and discuss the methods on which they rely. In Sect. 3 the relationship between and is discussed, depending on the method implemented. We explain how these geodetic data are taken into account in Sect. 4. We present our results in Sect. 5, and discuss them in the last part. We investigate how the use of a geodetic determination of the variable geopotential can influence the precession-nutation results, considering first the precession alone, and second the periodic contribution.
In the whole study, the time scale for t is TT Julian centuries since J2000, which will be denoted cy.
This section investigates the theoretical effect of the variations in the Earth's dynamical flattening on the precession expressions.
Figure 1: Angles and for the precession of the equator: is the mean equinox of the date and is the equinox of the epoch J2000.0. | |
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The two basic angles
and
(see Fig. 1) for the precession of the equator are
provided by the following differential equations (see Eq. (29) of Williams (1994) or Eq. (24) of Capitaine et al. (2003)):
(5) | |||
Table 1: Comparison between constants used for different determinations of the dynamical flattening (H): (1) the precession rate in longitude (); (2) the speed of the general precession in longitude (p_{1}); (3) the geodesic precession (p_{g}); and (4) the obliquity of the ecliptic at J2000.0 ( ). The observational value actually used for each study is written in bold.
We can write r_{0} as:
A major problem consists in choosing the constant value of H. Indeed, depending on the authors, it differs by about 10^{-7} (Table 1). This is due to the different measurements and models implemented (see Fig. 1 of Dehant & Capitaine 1997; Fig. 5 of Dehant et al. 1999). On the one hand the optical measurements give values of the general precession in longitude p_{A} referred to the ecliptic of date, whereas VLBI gives measurements relative to space. On the other hand, the various constants and models used for obtaining the value for H from a measured value (optical, Lunar laser ranging or VLBI) are different depending on the study considered (see Eq. (7)).
Classically, is developed in a polynomial form of t as: . In Table 1, we recall the different values used (i) for (i.e. the precession rate in longitude, ), directly obtained from VLBI measurements, and (ii) for p_{1} which is the observationally determined value of precession in the optical case: (Lieske et al. 1977).
The computation of the IAU2000 precession-nutation model by Mathews et al. (2002) is based on a new method which uses geophysical considerations. They adjust nine Basic Earth Parameters (BEP), including the Earth dynamical flattening H.
Based on the paper by Capitaine et al. (2003), denoted hereafter P03, we use differential Eq. (1) in which H has been replaced by (using Eqs. (2)-(4) and (7)). We start from the P03 initial values for the variables , , , and p_{A}, that are represented as polynomials of time and rely on the numerical values given in Table 2. We solve Eq. (1) together with the other precession equations (e.g. see Eqs. (26) and (28) of P03) with the software GREGOIRE (Chapront 2003) that can process Fourier and Poisson expressions. We iterate this process until we obtain a convergence of the solution.
From the geodetic C_{20} variation series we can derive the corresponding variations of the dynamical flattening H. Indeed, knowing that
,
in the case of a rigid Earth, we can write (see Lambeck 1988):
But the Earth is elastic, so let us consider small variations of H, C_{20} and the third principal moment of inertia of the Earth (C being its constant part and c_{33} its variable part). Then we obtain:
Table 2: Numerical values used in this study. H, and are integration constants.
The coefficient is usually obtained from the H and J_{2} values (see Eq. (8)). In order to determine the constant part of H, we can use (i) the , M and C values; or (ii) the Clairaut theory (see Table 3).
First, recall the Earth geometrical flattening :
In Table 3 we compare the various H values obtained. We denote (i) H^{*} the value obtained with the Clairaut method and (ii) H^{**} the value obtained using directly the , C and M values. Both are computed with Eq. (8) and a value for J_{2} of . Note that in contrast, IAG or MHB values (usually used) are determined from astronomical precession observations and can be used to compute the value. We can add that the differences with come from (i) the hydrostatic equilibrium hypothesis in Clairaut's theory for the value H^{*}; and (ii) the poorly determined , C and M values, for the value H^{**}. This will introduce errors in the determination, which we will study in Sect. 3.3.
In the following, we will use the value determined with the Clairaut theory, noted with a (*) in Table 3, which corresponds to a value for H of: .
Table 3: Comparison between different values of the coefficient and of the constant part for H: (1) IAG values (Groten 1999) - (2) MHB values (Mathews et al. 2002) - (3) Constant part H^{**} obtained from Eq. (8) using the M, and C IAG values - (4) Method of "Clairaut'' (Sect. 3.2), assuming hydrostatic equilibrium. The third and fourth methods use a constant part for of in Eq. (8) (i.e. ). The sense of the computation is indicated by the arrows.
We can estimate the error that the use of the Clairaut theory introduces into the
results. Indeed, if we consider the MHB value as the realistic H value (see Table 3), the relative error made is:
The geodetic data used are the time series (variable part) of the spherical harmonic coefficient C_{20} of the geopotential, obtained by the GRGS (Groupe de Recherche en Géodésie Spatiale, Toulouse) from the precise orbit determination of several satellites (like LAGEOS, Starlette or CHAMP) from 1985 to 2002 (Biancale et al. 2002). The combination of these satellites allows the separation of the different zonal geopotential coefficients, more particularly of J_{2} and J_{4}. This series includes (i) a model part for the atmospheric mass redistributions (Chao & Au 1991; Gegout & Cazenave 1993) and for the oceanic and solid Earth tides (McCarthy 1996); and (ii) a residual part (see Fig. 2) obtained as difference of the space measurements with respect to a model. These various changes in the Earth system are modelled as variations in the standard geopotential coefficient C_{20} and we note the different contributions , , and , respectively.
Figure 2: Normalized residuals ( top: raw residuals, bottom: filtered residuals, where the high frequency signals have been removed): non-modelled part of the harmonic coefficient of the Earth gravity field. | |
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Earlier studies already took into account the effect of the secular variation of C_{20} on the precession of the equator. Such a secular variation is attributed to the post-glacial rebound of the Earth (Yoder et al. 1983), which reduces its flattening. Williams (1994) and Capitaine et al. (2003) considered a J_{2} rate value of /cy. Using the numerical value of Table 2 for the first order contribution ( ) to the precession rate r_{0}, which is directly proportional to J_{2}, the contribution of the J_{2} rate to the acceleration of precession is about -0.014 ''/cy^{2}, giving rise to a -0.007 ''/cy^{2} contribution to the t^{2} term in the expression of .
Since 1998, a change in the secular trend of the J_{2} data has been reported (Cox & Chao 2000). This change can be seen in the series of residuals (see Fig. 2). An attempt to model this effect, with oceanic data, water coverage data and geophysical models, has been investigated by Dickey et al. (2002). Using the residuals of the GRGS, we can estimate a secular trend for from 1985 to 1998 (see Fig. 3). We find a J_{2} rate of the order of: /cy, which gives a change of about -0.006''/cy^{2} in the t^{2} term of the polynomial development of the precession angle .
As this secular trend is not the same in the total data span, we will also model the long term variations in the C_{20} residual series with a periodic signal. Such a long-period term in the J_{2} residual series may come from mismodelled effects, particularly from the 18.6-yr solid Earth tides. We will make such an assumption and adjust for the period 1985-2002, a secular trend and a long-period term in the residual series (see Sect. 4.3).
However, it should be noted that a secular trend for J_{2}, of the order of /cy, is more consistent with long term studies of the Earth rotation variations by Morrison & Stephenson (1997), based upon eclipse data over two millennia (they found /cy).
Figure 3: J_{2} GRGS residuals ( top: raw residuals, bottom: filtered residuals, where the high frequency signals have been removed): estimation of the linear trend, from 1985 to 1998. | |
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Figure 4: Normalized atmospheric ( top: raw data, bottom: filtered data, where the high frequency signals have been removed): atmospheric modelled part of the harmonic coefficient of the Earth gravity field, obtained with ECMWF pressure data. | |
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Figure 5: Normalized oceanic ( top: raw data, bottom: filtered data, where the high frequency signals have been removed): oceanic-tide-modelled part of the harmonic coefficient of the Earth gravity field; IERS Conventions 1996. | |
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Figure 6: Normalized solid tides ( top: raw data, bottom: filtered data, where the high frequency signals have been removed): solid-Earth-tide-modelled part of the harmonic coefficient of the Earth gravity field; IERS Conventions 1996. | |
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The geophysical models that have been previously subtracted from the C_{20} data (i.e. atmospheric, oceanic and solid Earth tides effects) must be added back to these data in exactly the same way they had been subtracted to reconstruct the relevant geophysical contributions.
For each contribution we give the associated potential U at the point (limited to the degree 2 and order 0) that we identify with the Earth gravitational potential. Hence, we obtain the coefficient contribution of each geophysical source.
(22) |
(23) |
(25) |
(26) |
(28) |
(29) |
Figure 7: Normalized total : top is the total series including atmospheric, oceanic tides and solid earth tides effects and the residuals; bottom is the total series without the solid earth tides effect. | |
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Equation (11) allows us to transform the geodetic temporal variations into the dynamical flattening variations . They can then be introduced into the precession Eq. (1), replacing H with (Eqs. (2)-(4) and (7)) and using the process already described in Sect. 2.3.
It is generally considered that VLBI observations of the Earth's orientation in space are not sensitive to the atmospheric and oceanic contributions to the variations in C_{20} (de Viron 2004). However the amplitudes of these effects have been evaluated in Table 11 for further discussion and in any case we can notice that they have a negligible effect on precession.
The analytical and semi-analytical approach to solving the precession-nutation equations provides polynomial developments of the and quantities. The data are then considered as a linear expression plus Fourier terms with periods derived from a spectral analysis (18.6-yr, 9.3-yr, annual and semi-annual terms) (see Tables 4-7). Note that the phase angles used for adjusting the periodic terms are those of the corresponding nutation terms. This implies changes in the development of the equatorial precession angles (, ), which we describe in the next section.
For the residual contribution of ,
we will consider (i) an adjustment of a secular trend over the interval from 1985 to 1998 (see Table 6 and Eq. (30)), and (ii) an adjustment of a secular trend plus a 18.6-yr periodic term (see Table 5 and Eq. (31)), both added to the seasonal terms. The fit (i) of the secular trend gives:
We must recall that these adjustments have been made together with the fit of annual and semi-annual terms. In contrast, the higher frequency terms appearing in the data have been filtered and we therefore did not take into account other contributions, as for example the diurnal effects of the geophysical contributions in .
Table 4: Summary of the constant parts for H and C_{20} (Constant part + Permanent tide) used in this study.
Table 5: Adjustment of periodic terms in the contributions, for the data span 1985-2002 for various geophysical sources (atmospheric , oceanic tides and solid earth tides , as well as the residuals - Units are in 10^{-10} rad.
On the basis of the models fitted to the time series of in the previous section, obtained with geodetic series, we investigate the influence of these geodetic data on the precession angle developments. First, we evaluate the effect of the secular trend considered in the residual series. Second, we report on the influence of each geophysical contribution, on the influence of the residuals and on that of the total contribution. Finally we focus on the periodic effects resulting from the various contributions.
We have already mentioned that the influence was taken into account in previous precession solutions (Williams 1994; Capitaine et al. 2003) (see Sect. 4.1). But depending on the value adopted, the polynomial development of the precession angle is different. Indeed, if we take /cy like in our study, or /cy like in Capitaine et al. (2003), the contribution in varies by about 1.5 mas/cy^{2} (see Table 8). So we must carefully take into account this J_{2} rate. Furthermore, (i) we already noticed that such a secular trend has been recently discussed because of the change in this trend in 1998 (see Fig. 2); and (ii) the uncertainty in this secular trend, derived from space measurements of J_{2}, is significant. Therefore we can conclude that until there is a better determination of the J_{2} rate, the accuracy of the precession expression is limited to about 1.5 mas/cy^{2}.
Table 6: Specific adjustment of the residual series ( ), from 1985 to 1998. The secular trend is considered as in Eq. (30) - Units are in 10^{-10} rad.
Table 7: Adjustment of the total series of ( ), from 1985 to 2002 - Units are in 10^{-10} rad.
Table 8: Influence of on the polynomial development of (more particularly on the t^{2} and t^{3} terms): (1) IAU2000 (Mathews et al. 2002); (2) P03 (Capitaine et al. 2003); and (3) Same computation as in P03 but with other values. The J_{2} secular trend estimation based on our C_{20} residuals series is: /cy.
Table 9: Polynomial part of the and developments (units in arcseconds): comparison of (1) IAU2000 (Mathews et al. 2002) - (2) P03 (Capitaine et al. 2003) - (3) Differences of Geod04 (this study) with respect to P03, considering all the contributions for (Table 7, ) - (4) Differences of Geod04 with respect to P03, obtained with a H constant part different from , but not used in the following (see Table 3 for the H^{*} and H^{**} constant values).
Table 10: Polynomial part of the development (units in arcseconds) for various sources used in our study, with respect to P03: comparison of (1) P03 (Capitaine et al. 2003) - (2) Difference between P03 and Geod04 (i.e. the effect of the total ) - (3) Difference between P03 and the effect of the residuals.
Table 11: Fourier part of the development, depending on the contribution considered for the periodic effect (units in as).
First, we can compare the polynomial part of our solution Geod04 for the precession angles, based on the constant part of H and on its variable part provided by expression (31), with previous precession expressions (IAU2000 and P03) (see Table 9). The differences larger than one as concern the precession angle and more particularly its t^{2} and t^{3} terms. The differences (of 7 mas and 2 as, respectively) with respect to P03 are due to considering or not considering the effect. Actually, P03 includes a J_{2} secular trend, whereas Geod04 includes instead a 18.6-yr periodic term (see (3) in Table 9 or (2) in Table 10). Comparing Geod04 with the IAU2000 precession (which does not consider the J_{2} rate) shows differences of 0.6 mas and 5 as in the t^{2} and t^{3} terms, respectively. This results from the improved dynamical consistency of the Geod04 solution (based on the P03 precession equations) with respect to IAU2000. Note that such results regarding the t^{2} and t^{3} terms will not be affected if changes of the order of 1 mas/cy in the precession rate would occur in an updated P03 solution.
Second, we can evaluate the differences introduced in the (and ) polynomial development by the use of a constant part for H determined with the geodetic J_{2} (as used in Geod04-H* and Geod04-H**) instead of the determined by VLBI and used in Geod04. Table 9 shows that the differences are very large, but it should be noted that using J_{2} for deriving H suffers from the too large errors introduced by the mismodelled .
On the basis of the adjustments made in Sect. 4.3 for the different contributions, we estimate here the periodic effects appearing in the expressions of the precession angles. We can focus on the Fourier terms in the precession angle, which are the most sensitive to the effects. The corresponding results are presented in Table 11.
This study was based on new considerations: the use of a geodetic determination of the variable geopotential to investigate its influence on the developments of the precession angles. The major effect on the precession is due to the J_{2} secular trend which implies an acceleration of the precession angle. But for the moment, the available time span for J_{2} satellite series is not as long as we need to determine a reliable value. The J_{2} secular trend estimation based on our C_{20} residuals series from 1985 to 1998 is: /cy. The accuracy of the precession expression is limited to about 1.5 mas/cy^{2} due to the uncertainty in this J_{2} rate value.
Then, we can notice that the main periodic effect is due to the 18.6-yr periodic term in due to solid Earth tides. But we must say that computing the with satellite positioning observations requires making some assumptions on the geophysical contributions to , for instance from atmospheric pressure, and oceanic or solid Earth tides. Actually, models are used, but they are not perfect and we may have some errors. So the residuals obtained may be affected by these errors, which is why the total contributions (residuals observed + models assumed) constitute a better series to evaluate the effects on the precession angles. This introduces Fourier terms into the development (as and as in cosine and sine respectively; see Table 11) that we should compare to the MHB2000-nutations. Indeed, the different terms of the total (or ) contributions have same periods as the ( , ) nutations. This implies that there is some coupling between the observed effects and the nutations, which may not have been included in the MHB2000-nutations.
In the future, we will be able to compare the J_{2} data with geophysical models and data, in order to have better ideas on the different contributions and on the secular trend. We will also be able to proceed to numerical study of this problem, and to implement a refined and more realistic Earth model.
Acknowledgements
We are grateful to V. Dehant for helpful advice and information. We thank C. Bizouard, O. de Viron, S. Lambert, and J. Souchay, for valuable discussion and J. Chapront for providing the well documented software GREGOIRE. We also thank the referee for valuable suggestions for improving the presentation of the manuscript.