The second of Eq shows that the coefficient of fifth-order term for the general precession in longitude.

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The IAU Working Group on Precession and the Equinox looked at several solutions for replacing the precession part of the IAU 2000A precession–nutation model, which is not consistent with dynamical theory. These comparisons show that the (Capitaine et al., Astron. Astrophys., 412, 2003a) precession theory, P03, is both consistent with dynamical theory and the solution most compatible with the IAU 2000A nutation model. Thus, the working group recommends the adoption of the P03 precession theory for use with the IAU 2000A nutation. The two greatest sources of uncertainty in the precession theory are the rate of change of the Earth’s dynamical flattening, J2, and the precession rates (i.e. the constants of integration used in deriving the precession). The combined uncertainties limit the accuracy in the precession theory to approximately 2 mas cent−2. Given that there are difficulties with the traditional angles used to parameterize the precession, zA, ζA, and θA, the working group has decided that the choice of parameters should be left to the user. We provide a consistent set of parameters that may be used with either the traditional rotation matrix, or those rotation matrices described in (Capitaine et al., Astron. Astrophys., 412, 2003a) and (Fukushima Astron. J., 126, 2003). We recommend that the ecliptic pole be explicitly defined by the mean orbital angular momentum vector of the Earth–Moon barycenter in the Barycentric Celestial Reference System (BCRS), and explicitly state that this definition is being used to avoid confusion with previous definitions of the ecliptic. Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 10 JAN 2006 2. REPORT TYPE N/A 3. DATES COVERED 4. TITLE AND SUBTITLE Report of the International Astronomical Union Division 1 Working Group on Precession and the Ecliptic 5a. CONTRACT NUMBER

On 21 July 1969, the first lunar reflector was placed on the Moon surface by Apollo 11 mission. Since this date 4 more reflectors have been deployed and, today, the Lunar Laser Ranging (LLR) is still operating in USA and France. The complete set of LLR observations covers a time interval longer than 30 years and the accuracy of measurements is regularly improved. This technique has brought remarkable results in lunar science, solar system dynamics, geophysics and fundamental physics. In 1998, the Paris Observatory Lunar Analysis Center (POLAC) has performed analysis of LLR observations. In this contribution we emphasize some results obtained in 2002 such as the orientation of the dynamical ecliptic, the correction to the IAU 1976 value of the precession constant in longitude and the improvement of the tidal acceleration of the Moon. Lunar laser telemetry consists in determining the round-trip travel time of the light between a transmitter on the Earth and a reflector on the Moon, which is an equivalent measurement of the distance between these two points. Neil Armstrong put down the first reflector array in the Sea of Tranquillity in July 1969 and a few weeks later McDonald Observatory succeeded in detecting photons returned from a laser pulse sent to the Moon. Afterwards, more reflector arrays have been placed by two American Apollo missions in 1971 (Apollo 14 and Apollo 15), and two Soviet automatic missions, Luna 17 (1970) and Luna 21 (1973) which carried French-built reflectors, Lunakhod 1 and 2. Except Lunakhod 1 from which the signal has been lost, the 4 other reflectors are still operating normally (Fig. 1). Fig. 1. Distribution of the reflectors on the lunar surface The Apollo arrays consist of 3.8-cm diameter corner cubes mounted on an aluminium panel (100 for Apollo 11 and 14 and 300 for Apollo 15). The Lunakhod arrays have 14 triangularly faced corner cubes of 11-cm edge. The design of reflector arrays is straightforward: each corner cube reflects incident light back to its point of origin (Fig. 2).

We consider 4 libration models : 3 numerical models built by JPL (ephemerides for the libration in DE245, DE403 and DE405) and an analytical model improved with numerical complements fitted to recent LLR observations. The analytical solution uses 3 angular variables (p1, p2, τ) which represent the deviations with respect to Cassini’s laws. After having referred the models to a unique reference frame, we study the differences between the models which depend on gravitational and tidal parameters of the Moon, as well as amplitudes and frequencies of the free librations. It appears that the differences vary widely depending of the above quantities. They correspond to a few meters displacement on the lunar surface, reminding that LLR distances are precise to the centimeter level. Taking advantage of the lunar libration theory built by Moons (1984) and improved by Chapront et al. (1999a) we are able to establish 4 solutions and to represent their differences by Fourier series after a numerical substitution of the gravitational constants and free libration parameters. The results are confirmed by frequency analyses performed separately. Using DE245 as a basic reference ephemeris, we approximate the differences between the analytical and numerical models with Poisson series. The analytical solution improved with numerical complements under the form of Poisson series is valid over several centuries with an internal precision better than 5 centimeters. 1. PRESENTATION OF THE MODELS • Cassini’s laws. The lunar libration is characterized by small oscillations around an equilibrium position governed by Cassini’s laws: (i), the rotation period of the Moon is identical to its circulation period in the orbital motion; (ii), the inclination of the lunar equator on the ecliptic is a constant; (iii), the secular motions of the nodes N and N ′ on the ecliptic of the orbital plane and the lunar equator are identical (see Fig. 1). • The variables. A selenodesic system of axes (ξ, η, ζ) along the principal moments of inertia (A, B, C) is connected with the ecliptic system (X, Y , Z) by 3 Euler’s angles (φ, ψ, θ) as shown on Fig. 1. Three position angles denoted by (p1, p2, τ) express the small oscillations around the equilibrium position. They are referred to Euler’s angles by the relation: p1 = sinφ sin θ; p2 = cosφ sin θ; τ ∗ = φ+ ψ or τ = τ − w1 − 180 where w1 is the mean longitude of the Moon. p1 and p2 are the components of the unit vector pointing towards the mean pole of the ecliptic of date on the two lunar equatorial principal axes of inertia; τ is the libration in longitude. In the analytical theory the position variables are (p1, p2, τ). In the 3 JPL lunar ephemerides Euler angles are used instead.

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 J(2) 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.

Three independent high precision solutions for precession were published in 2003 that provide expressions consistent with the IAU 2000A precession-nutation model (Mathews et al. 2002) and offer a possible replacement for the precession component of IAU 2000A, with improved dynamical consistency and a better basis for future improvement. Each is based upon an improved model for the precession of the ecliptic and, with respect to the IAU 1976 precession, they all provide higher-degree terms in the polynomials for the precession angles of the equator. This paper compares the expressions for the basic parameters of the above solutions for precession both of the ecliptic and the equator and investigates the possible physical and computational reasons for their differences. This leads to a realistic evaluation of the accuracy of the solutions and provides estimated deficiencies in them. These studies have identified expressions for the ecliptic precession quantities that are accurate to about 0.05 mas/cy over a two-millennium interval centered on J2000 instead of the few mas/cy accuracy of the current IAU model. They have also provided the theoretical and experimental basis for future improvements in the precession of the equator.

On the basis of the semi- analytical theory ELP, a new solution has been built that makes use of the planetary perturbations MPP01 constructed by P. Bidart. This new solution, called ELP / MPP02, is an analytical solution that contains all the perturbations to represent the lunar motion. The level of truncation for the series is the centimeter. This limit induces analytical developments of a reasonable dimension. ELP / MPP02 is compared to the JPL ephemerides DE405 and DE406. After fitting the constants and the reference frame to DE405, over a few centuries, numerical comparisons with DE405 show a significant improvement in precision, in particular for the radius vector. Over the long range of several millennia, comparisons with DE406 show an improvement in the long periodic perturbations due to planets. On the time interval of one century around J2000 we added to ELP / MPP02 numerical complements rho(405) based on the differences with DE405 in such a way that ELP / MPP02+ rho(405) has the precision of the numerical integration. Using this new ephemeris to analyse LLR data provided since 1970, we build the solution ELP / MPP02( LLR) fitted to LLR observations. Various approximations have been also tested on the residuals DE406 - ELP / MPP02. We first determined corrections to the secular variations of the mean longitude, longitudes of the node and perigee under a simple polynomial form; they reduce the differences between ELP / MPP02 and DE406 to less than 3 arcsec in longitude and latitude and 2 km in distance over the whole time interval of DE406 [- 3000; + 3000]. Next, the differences themselves between the two solutions are approximated over 2 millennia with Poisson series in a pseudo analytical form similar to ELP / MPP02. We reduce the residuals to less than 0.03 arcsec in longitude and latitude and 30 m in distance.

Expressions for the position of the Celestial Intermediate Pole (CIP) and the Celestial Ephemeris Origin (CEO) in the Geocentric Celestial Reference System (GCRS) have been computed using the IAU 2000A precession-nutation ? .T hese expressions are for use in the new transformation between the GCRS and the International Terrestrial Reference System (ITRS) which is recommended by IAU Resolution B1.8. Various comparisons and numerical checks have been performed between the classical and the new transformations based on the IAU 2000A precession-nutation. These comparisons revealed necessary improvements to be applied to the classical form of the transformation in order to achieve the required level of accuracy. Once these improvements are applied, the consistency between the positions of the CIP in the GCRS corresponding to the classical and the new transformations is at a level of a few microarcseconds after one century. This work has demonstrated that the new method, in addition to providing an explicit separation between precession-nutation of the equator from Earth rotation, is more simple, compact and direct than the classical one, achieving accuracies at the level of a few microarcseconds with greatly reduced scope for accidental misuse. The resulting expressions for X, Y and s have been included in the IERS Conventions 2000. References for the numerical expressions are provided in Appendix C.

A new precession-nutation model for the Celestial Intermediate Pole (CIP) was adopted by the IAU in 2000 ( Resolution B1.6). The model, designated IAU 2000A, includes a nutation series for a non-rigid Earth and corrections for the precession rates in longitude and obliquity. The model also specifies numerical values for the pole off sets at J2000.0 between the mean equatorial frame and the Geocentric Celestial Reference System ( GCRS). In this paper, we discuss precession models consistent with IAU 2000A precession-nutation (i.e. MHB 2000, provided by Mathews et al. 2002) and we provide a range of expressions that implement them. The final precession model, designated P03, is a possible replacement for the precession component of IAU 2000A, offering improved dynamical consistency and a better basis for future improvement. As a preliminary step, we present our expressions for the currently used precession quantities zeta(A); theta(A); z(A), in agreement with the MHB corrections to the precession rates, that appear in the IERS Conventions 2000. We then discuss a more sophisticated method for improving the precession model of the equator in order that it be compliant with the IAU 2000A model. In contrast to the first method, which is based on corrections to the t terms of the developments for the precession quantities in longitude and obliquity, this method also uses corrections to their higher degree terms. It is essential that this be used in conjunction with an improved model for the ecliptic precession, which is expected, given the known discrepancies in the IAU 1976 expressions, to contribute in a significant way to these higher degree terms. With this aim in view, we have developed new expressions for the motion of the ecliptic with respect to the fixed ecliptic using the developments from Simon et al. ( 1994) and Williams ( 1994) and with improved constants fitted to the most recent numerical planetary ephemerides. We have then used these new expressions for the ecliptic together with the MHB corrections to precession rates to solve the precession equations for providing new solution for the precession of the equator that is dynamically consistent and compliant with IAU 2000. A number of perturbing effects have first been removed from the MHB estimates in order to get the physical quantities needed in the equations as integration constants. The equations have then been solved in a similar way to Lieske et al. ( 1977) and Williams ( 1994), based on similar theoretical expressions for the contributions to precession rates, revised by using MHB values. Once improved expressions have been obtained for the precession of the ecliptic and the equator, we discuss the most suitable precession quantities to be considered in order to be based on the minimum number of variables and to be the best adapted to the most recent models and observations. Finally we provide developments for these quantities, denoted the P03 solution, including a revised Sidereal Time expression.

Expressions for the position of the Celestial Intermediate Pole (CIP) and the Celestial Ephemeris Origin (CEO) in the Geocentric Celestial Reference System (GCRS) have been computed using the IAU 2000A precession-nutation. These expressions are for use in the new transformation between the GCRS and the International Terrestrial Reference System (ITRS) which is recommended by IAU Resolution B1.8. Various comparisons and numerical checks have been performed between the classical and the new transformations based on the IAU 2000A precession-nutation. These comparisons revealed necessary improvements to be applied to the classical form of the transformation in order to achieve the required level of accuracy. Once these improvements are applied, the consistency between the positions of the CIP in the GCRS corresponding to the classical and the new transformations is at a level of a few microarcseconds after one century. This paper summarizes the main points of this work, which will be described in more detail in a paper in preparation (Capitaine et al. 2002).

An analysis of Lunar Laser Ranging (LLR) observations from January 1972 until April 2001 has been performed, and a new solution for the lunar orbital motion and librations has been constructed that has been named S2001. With respect to prior solutions, improvements in the statistical treatment of the data, new nutation and libration models and the addition of the positions of the observing stations to the list of fitted parameters have been introduced. Globally, for recent observations, our rms (root mean square error) is within 2 to 3 centimeters in the lunar distance. Special attention has been paid to the determination of the correction to the IAU76 luni-solar constant of precession, and the value of the secular acceleration of the Moon's longitude due to the tidal forces. The main results are:correction to the constant of precession: Deltap = -0.302+/-0.003"/cy,tidal acceleration of the lunar longitude: Gamma = -25.858+/-0.003"/cy(2).The positions and velocities of the stations have also been determined. The results are consistent with the ITRF2000 determinations from SLR observations. The lunar theory ELP is referred to a dynamical system and introduces the inertial mean ecliptic of J2000.0. The positioning of the reference system of the theory with respect to ICRS is performed (and also with respect to some useful JPL numerical integrations). Finally the orientation of the celestial axes with respect to the ICRS reference system has been derived as well as the offsets of the Celestial Ephemeris Pole.